t J 



vj^O 




Class. 
Book 



fui 



ELEMENTS 

OF THE 

DIFFERENTIAL AND INTEGRAL 

CALCULUS 

WITH 

Examples and Applications 



JAMES Mi TAYLOR, A.M., LL.D 

PROFESSOia OF MATHEMATICS 
COLGATE UNIVEKSITY 



REVISED EDITION 



BOSTON, U.S.A. 

GTNN & COIMTANV. PUr>MSllKHS 

dTIjc 9[tI)cn<Tum J3icd6 



Copyright, 1884, 1891 
By JAJMES M. TAYLOR 



COPYEIGHT, 1898 
By JAiVIES J\I. TAYLOK 



ALL TtTPxHTS RT:SEKVED 



By Gxohang-e 
Army S^ Havy Club' 

JUN 2 2 mo 






PREFACE TO THE REVISED EDITION. 



5j«^C 



In this revision an attempt has been made to present in 
their unity the three methods commonly used in the Calculus. 
The concept of Rates is essential to a statement of the prob- 
lems of the Calculus ; the principles of Limits make possible 
general solutions of these problems, and the laws of Infini- 
tesimals greatly abridge these solutions. 

The Method of Rates, generalized and simplified, does not 
involve "the foreign element of time." For in measuring 
and comparing the rates of variables, the rate of any variable 
may be selected as the unit of rates, dy /dx is the x-rate of y, 
or the ratio of the rate of y to that of x, according as the rate 
of X is or is not the unit of rates. 

The proofs of the principles of differentiation by the 
Method of Rates, and the numerous applications to geometry, 
mechanics, etc., found in Chapter II, render familiar the 
problem of rates before its solution by the ^lethod of Limits 
or Infinitesimals is introduced. 

In Cliapter III, by proving that It (A///A.r) = r/// wAr, tlie 
problem of rates is reduced to the i)r()blem of iiuding the 
limit of the ratio of infinitesimals. 

Tlie Theory of Infinitesimals is that jiart of Ww ThiH>rv of 
Limits which treats of variables Jiari)^/ zero as their common 
limit. In approaching its limit, an intinitesiuial ]>asses 
through a series of linitely small vahu^s befm-e it reaches 
infinitely small values. Litinitesimals can be tlivided into 
orders, and their laws can be established anil apj>lied when 



iv PREFACE. 

they are finitely small as well as when they are infinitely 
small. Hence, in the study of infinitesimals it is not neces- 
sary to determine that indefinite boundary between the finitely 
small and the infinitely small. Any small quantity becomes 
an infinitesimal wdien it begins to approach zero as its limit, 
not when it reaches any particular degree of smallness. A 
quantity, however small, which does not approach zero as its 
limxit is not an infinitesimal. If it is recognized that the 
essence of infinitesimals lies in their having zero as their limits 
rather than in their smallness, the study of them ceases to be 
mystical, obscure, and difiicult. 

Again, the concept of a limit as a constant whose value the 
variable never attains removes the necessity of studying the 
anatomy of Bishop Berkeley's ^' ghosts of departed quanti- 
ties." Infinitesimals never equal zero and should not be 
denoted by the zero symbol. This distinction between infini- 
tesimals and zero involves that between infinites and a/0. 

The much-abused form 0/0 cannot arise in the Calculus or 
elsewhere from any principle of limits ; a distinctive service 
of the Theory of Limits is that it enables us to evaluate any 
determinate expression when it assumes this or any other 
indeterminate form. 

Those who prefer to study the Calculus by the Method of 
Limits or Infinitesimals alone can omit the few demonstra- 
tions in Chapter II, which involve rates, and substitute for 
them the proofs by limits or infinitesimals in Chapter III. 

To meet an increasing demand for a short course in differ- 
ential equations, a chapter has been devoted to that subject. 

A table of integrals arranged for convenience of reference 
is appended. 

Throughout the work, as in previous editions, there are 
numerous practical problems from mechanics and other 
branches of applied mathematics which serve to exhibit the 
usefulness of the science, and to arouse and keep alive the 
interest of the student. 



PREFACE, V 

At the option of the teacher or reader, Chapters I and II 
of the Integral Calculus can be read after completing Chapters 
I and II of the Differential Calculus ; also many of the 
numerous examples and problems may be omitted. 

The author takes this opportunity of expressing his grati- 
tude to the friends who by encouragement and suggestions 
have aided him in this revision. 

James M. Taylor. 
Hamiltox, N. Y., 1898. 



OONTEITTS. 



>>!^c 



PART I. — DIFFERENTIAL CALCULUS. 



Chapter I. 



Functions. Rates. Differentials. 

Sections Pages 

1-5. Classification of variables and functions 1,2 

6-8. Notation of functions. Continuity. Increments . . . 3-5 

9, 10. Uniform change. Measure of rates 0, 7 

11. Differentials 8 

12,13. cZ?//dx, a rate or a ratio of rates 

14. Definition of the Differential Calculus 10 

Chapter II. 

Differentiation. Applications. 

15-28. Differentiation of algebraic, logarithnnc, and cxiionential 

functions 11-20 

21), 30. Derivatives. Velocity 21-23 

31-33. Tangent, dy / dx — slope 24 

34, 35. Equations of tangents and nornu\ls 25 

36. Values of subl., subn., (an., and norm 2(>-20 

37-44. DilTerentialiiHi of sin </, cos H, etc ;U> ;VJ 

45-52. DifferiMitiation of sin-M<, cos M<, etc. ;>•> -m 

JNIiscellancous exaniples ... 38, .U) 



Vlll 



CONTENTS. 



Chapter III. 



Problem of Rates Solved by Limits. 
Sections 

53-55. Limits. Notation. Lt (Ay/Ax) = dy/clx 

56, 57. Lt {Ay /Ax), how found. Derivative as a limit 

58-60. Infinitesimals. Infinites. Lt {Ay /Ax) = slope 

61,62. Theorems concerning limits 

63. Subt., subn., tan., and norm., in polar curves 

64-66. Orders of infinitesimals. Notation 

67-70. Theorems concerning infinitesimals 

71. Rule for differentiating a function 

72, 73. Orders of infinites, con . o = 

74. The symbol op 

75, 76. Use of infinitesimals. Limits in position 



Pages 

40-42 

43 

43, 44 

45 

46,47 

48, 49 

50 

51-53 

54 

55 

56 



Chapter IV. 

Successive Differentiation. 

77. Successive differentials 57 

78-80. Successive derivatives 58-01 

81, 82. Leibnitz's theorem. Acceleration 62-64 



Chapter V. 



Indeterminate Forms. 

83. Value of a function of x for x = a 65 

84. Indeterminate form 0/0 66 

85. Indeterminate form op /op 67 

86. Indeterminate forms ■ op, op — op 68 

87. Indeterminate forms 0'), 0^0, 1 ± 0^ 69 

88. Evaluation of derivatives of implicit functions .... 70 



CONTENTS. ix 

Chapter VL 

Expansion of Functions. 

Sections Pages 

89,90. Series. Expansion of a function . 71 

91-93. Taylor's and Maclaurin's theorems ...... 72-74 

94. Expansions of sin X and cos X 75 

95, 96. The exponential series. Second form of Rr ■• . 76 

97. The logarithmic series 77 

98. The binomial theorem . 78 

99. Failure of Taylor's and Maclaurin's theorems . . . 79-81 



Chapter VIT. 

Maxima and Minima. 

100. Definitions of maximum and minimum 82 

101-104. Theorems concerning maxima and minima of /x . . 82,83 

105-107. Methods of examining /x for critical values of x . . 84, 87 

Problems in maxima and minima 88-92 



Chapter VIII. 

Points of Inflexion. Curvature. Evolutes. 

108-110. Points of hitiexiou 93,94 

111-113. Curvature 95 

114. Circle of curvature 9(5 

115. /i' in ((>ruKs of .(■ and // 9(i, 97 

116. /.' in (vnns of /) and (' 98 

117. (\>-(>rdiuatos of ciMitfi' of cuvNaluro !K^ 

118,119. PropiMlii's of ovohiti's anil invohUrs . . 



)') 



120. K(iuations of ('\n>hitivs \00 lOi 



X CONTENTS. 

Chapter IX. 

Envelopes. Order of Contact. Osculating Curves. 

Sections Pages 

121-123. Family of curves. Envelopes 103, 104 

124. Equations of envelopes 104-106 

125. Different orders of contact 107 

126. Conditions of contact of the kih order 108 

127-130. Osculating curves 109-111 

Chapter X. 

Change of the Independent Variable. 

131,132. Change of the independent variable 112-114 

Chapter XI. 
Functions of Two or More Variables. 

1.33. Partial differentials and derivatives. 115 

134,135. Total differentials 116 

136. Derivative of an implicit function 117 

137, 138. Derivative of a function of a function 118 

139, 140. Successive partial differentials and derivatives . . 119-121 

141. Successive total differentials 122 

142. Expansion of/ (a: + /i, ?/ + A;) 122,123 

143,144. Maxima and minima of / (a:, ?/) 123-126 

Chapter XII. 

Asymptotes. Singular Points. Curve Tracing. 

145-148. Asymptotes to curves. Rectangular axes .... 127-131 

149. Asymptotes to polar curves 132 

150-153. Multiple points. Conjugate points 133-136 

154, 155. Symmetry. Curve tracing 137-142 



CONTENTS. xi 



PART IL — INTEGRAL CALCULUS. 



Chapter L 



Standard Forms. Direct Integration. 
Sections Pages 

166-158. Generalintegrals. Elementary principles . . . . 143,144 

159-163. Standard formulas. Direct integration .... 145-157 

Chapter IL 
Definite Integrals, Applications. 

164, 165. Definite and corrected integrals 158-160 

166. Geometric meanings of j (x) dx, I (x) cZx, 

and r0(x)(^x ! 161,162 

167, 168. Accelerated motion. Change of limits .... 163-166 

Chapter III. 

Integration of Rational Fractions. 

169-173. Decomposition and integration of fractions . . . 167-173 

Chapter IV. 
Integration by Rationalization. 

174. Rationalization by substitution 174 

175. Differentials involving (rt + 6.r) '"/ " 174, 17-'> 

176, 177. Differentials involving V±TM-^ax~-F7) .... 176, 177 

178. Differentialsof the fornix'" ((f + 6x")'''-^tZx . . . 178-180 

Chapter V. 

Integration by Parts. Reduction Formulas. 

179, 180. lii(,(«ora(i()ii by parts. Standard foriunlas ... ISl 1S4 

181-183. llediu'tion formulas fm- j X'" ((f + ^x")''(/.r . . . 1 80 189 



xii CONTENTS. 

Chapter VI. 

Integration of Trig-onometric Forms. 

Sections Pages 

184-186. Forms, j tan" It cZu, i seC'udu, | tan™ m sec" w du . 190,191 

187-191. Form, j sin™ m cos" m dzt 192-197 

192. Forms, f-^^^, f-J'^ 198 

J a + h cos u J a + sm u 

193. Integration of trigonometric forms by substitution. 199, 200 

194. Ce^^^mhxdx 201 

195. Integration by series 201, 202 

Chapter VII. 

Lengths and Areas of Curves. Surfaces and Volumes of Solids 
of Revolution. 

196-200. Lengtlie of curves. Catenary. Tractrix .... 203-207 

201,202. Areas bounded by curves 208-210 

203, 204. Surfaces and volumes of solids of revolution . . . 211-214 

205. Volumes generated by plane figures 215, 216 

Chapter VIIL 
Double and Triple Integration. Applications. 

206. Double and triple integration 217, 218 

207, 208. Areas by double integration 219, 220 

209. Surfaces by double integration 221, 222 

210. Volumes of any solids by triple integration . . . 223, 224 

211. Volumes of solids of revolution 225 

212, 213. General formulas for centre of mass 226 

214. Centre of mass of areas 227 

215. Centre of mass of curves 228-2.30 

216. Moment of inertia 231 



CONTENTS. xiii 

Chapter IX. 

Definite Integral as a Limit. Intrinsic Equations of Curves. 

Sections Pages 

217. Definite integral as a limit 232, 233 

218-220. Intrinsic equations of curves 234-236 

Chapter X. 
' Differential Equations. 

221-223. Differential equations. General solutions . . . 237 

224. Form, 0i xdx + <P2ydy = 238, 239 

225. Homogeneous equations 240 

226. Non-liomogeneous equations, linear in x and y . . 241, 242 

227. Exact differential equations 243, 244 

228. Integrating factor 245 

229, 230. Rules for finding the integrating factor .... 246-248 

231. Linear equations of the first order 249 

232. Equations reducible to the linear form 250 

233. Equations of the first order and nth degree . . . 251 

234. Equations of orders above the first 252-257 

APPENDIX. 

Table of integrals 259-2i)9 



ELEMENTS OE THE CALCULUS. 



3>S^C 



PART I. DIFFERENTIAL CALCULUS. 



CHAPTER L 

FUNCTIONS. RATES. DIFFERENTIALS. 

1. A variable is a quantity which is, or is supposed to be, 
continually changing in value. Variable numbers are usuallj^ 
represented by the final letters of the alphabet, as x, ?/, z. 

A constant is a quantity whose value is, or is supposed to 
be, fixed or invariable. Constant numbers may be individual 
or general. Individual constants are represented by figures ; 
general constants are usually represented by the first letters 
of the alphabet. 

For example, in the equation of the circle (x — 3)2 + (?/ — 4)- = f-, o 
and 4 are individual constants fixing the centre ; r is a general ctnistant 
denoting any radius; x and ?/ are variables denoting the coiirdinates of 
the moving point which traces the circle. 

2. Classification of variables. A variable whose value 
depends upon one or more other variables is called a de prud- 
ent variable, or {\. function of those variables. A variable whii'h 
does not depend ui)ou any otluu- variabh^ is calU^l an Indv- 
])ende7it variable. 

For example, x*' — /)-, sin .r, and U>g (.r — a) aro fun;Mioiis of ilu> iutlo- 
pcndent variable x. Again, x'' + '■>.(•// -|- //-, log (x- — i/-), and a' * ' are 
functions of the two vaviabU\s x and //. 



2 DIFFERENTIAL CALCULUS. 

3. Classification of functions. An algehraic function is 
one which without the use of infinite series can be expressed 
by the operations of addition, subtraction, multiplication, 
division, and the operations denoted by constant exponents. 
All functions which are not algebraic are called transcendental. 
Of these, the more common are : 

The exponential function y = a^ -^ and its inverse, the loga- 
rithmic function x = log„?/. 

The trigonometric functions ?/ = sin x, y = cos x, etc.; and 
the inverse-trigonometric functions x = sin~^y, x = cos~-^^, etc. 

4. Explicit and implicit functions. When an equation 
involving two or more variables is solved for any one of 
them, this one is said to be an explicit function of the others. 
When an equation is not so solved, any one of its variables is 
called an im2olicit function of the others. 

For example, in x- + y- = r-, eith er ?/ or a: is an implicit function of 
the othe r ; whil e in y = ±: Vr^ — x^, y is an explicit function of x, and in 
X = ± Vr- — ?/2, X is an explicit function of y. 

A function is said to be one-valued, two-valued, or n-valued 
according as for each value of its variable it has one value, 
two values, or n values. 

For example, y is a one-valued function in y = x^, a two- valued func- 
tion in y- =4px, and a three-valued function in y'^ + xy -^ x"^ = I. 

5. Increasing and decreasing functions. An increasing 
function is one which increases when its variable increases. 
Hence, it decreases when its variable decreases. 

A decreasing function is one which decreases when its 
variable increases. Hence, it increases when its variable 
decreases. 

For example, 5x and logx are increasing functions of x, while — 5x 
and 1/x are decreasing functions of x. 



NOTATION OF FUNCTIONS. 3 

6. Notation of functions. The symbols f(x), f'(x), F(x), 
cj) (x), and the like are used to denote different functions of x. 
Likewise, f(x, y), <^ (x, y) denote different functions of x and y. 

The equation y = f{x) expresses that y is an explicit func- 
tion of £c ; while f(x, y) = expresses that either y ot x is an 
implicit function of the other. 
,/ ^ In the same problem or discussion the symbol /( ) denotes 
the same function of one enclosed quantity as of another. 

For example, if f{x) = x^ + 2x-\-3; 

then in the same problem or discussion ^^ '^L^' 

/(2/) = Z/2 + 2 7/ + 3, 
and f{x + y) = (x + yY + 2 (x + ?/) + 3. 

Also, /(2), /(I), and /(O) denote respectively the values of /(x) for 
X = 2, X = 1, andx = 0. 

EXAMPLES. 

1. /(x) = 5x2-3x + 2; find/(?/), /(x + /O, /(3), /(O), /(- 2). 

2. /(^)=cos2^; find/(x),/(7/),/(0),/(7r/2),/(7r). 

3. f{z) = 2 ^4 — 2;3 + 1, and <p {z) = 1 z"- — d z + \ ; show that 

/(O) = (0), /(I) - (1), /(- 2) z: (- 2). 

4. /(x)zz(a + x)'«; timl/(2), /(/i - a), /(O), /(2). 

5. F{x) = e^ - e-^ ; show that I^^(3 x) - [F{x)Y -h 3 F{x). 

6- /(^) = logYTl' show that /{x) +/(//)-/ (^'_^-;^) • 

7. /(x + y) =: a^+" ; liiul /(x), /(^), /(;» 4- n). 

8. /(x, ?y)E:x'5 + r)X//- + 3//5 ; iiiul f{ni, u),/{-2, - 3). 

Of Iho f()lU)\viiis;- functions, which aiv iiuMvasiii-;- ami which arc dccroas- 
ino- functions of x ? 

9. 2'. 10. tanx. 11. - x'\ 12. -l/.r. 



4 DIFFERENTIAL CALCULUS. 

7. A continuous real variable is a variable whicli in passing 
from one real value to another passes successively through, all 
intermediate real values. 

A function f(x) is said to be real and continuous between 
X = a and x ^=h, if when x is real and changes continuously 
from a to h, f(x) is real and varies continuously from /(«) to 
f{h). In other words, f{x) is real and continuous between 
x = a and x = h when the real locus of y = f(x) between the 
points [a, /(«)] and [b, fip')'] is an unbroken curve. 

Some functions are real and continuous for all real values of 
their variables ; others are real and continuous only bet^^een 
certain limits. 

Eor example, the time since any past event varies continuously. The 
velocity acquired by a fa,lling body and the distance fallen are continuous 
functions of the time of falling. Most quantities in nature are continuous 
variables. Sin 6 and cos 6 are continuous functions for all real values of 6. 
Tan ^ is a continuous function of 6 between ^ = and 6 = ^/2, also 
between d=7C/2 and ^ = 37r/2; but when d passes through 7t /2 or 
Zti/1^ tan ^ leaps from -f co to -co; hence, tan is discontinuous for 
e = 7t/2 or S7t/2. 

In x'^ -i- y- = r^, y is a real and continuous function of x between 
X ■= — r and x = r. In a^y'^ — ft^x- = — a^^"^, y is a real and continuous 
function of x between x = — oo and x =— a, and between x = a and 
X = cc. Between x = — a and x = a, y is imaginary but continuous. 

The Calculus treats of coutiniious variables onl}^, or of 
variables between their limits of continuity. 

8. Increments. The amount of any change (increase or 
decrease) in the value of a variable is called an ijicrement. 
If a variable is increasing, its increment is j^ositive; if it is 
decreasing, its increment is negative. 

An increment of a variable is denoted by writing the letter 
A before it ; thus Aa:, Ay, and A/(cr) denote the increments of 
X, y, and f(x), respectively. 

If y =f(x), Ace and A?/ denote corresponding increments of 
x and y, and Ay = A/(cc). 



INCREMENTS OF VAKIABLES. 



Let PH be the locus of y =f{x) referred to the rectangular axes OX 
and OY. 

If when x= OA, Ax =AB, 
then Ay = BP' - AP = EP' ; 

if when x = OC, Ax = CF, 

Ay = FH- CD=-ND. 

In the last case Ay is negative, 
but it is properly called an incre- 
ment, since it is what must be added 
to the first value CD to produce the 
second FH. 

When X = OA = x', 

when X = OB = x' + Ax, 




hence, when 



Af(x) 



fix): 
: BP' 



AP=f{x'); 

BP' =/(x' + Ax); 

^P = f(x' + Ax)-f(x'). 



-^ ri 



EXAMPLES. 

1. The ratio of A (ax + h) to Ax is the constant a. 

Let y — ax + h. (1) 

Let x' and ?/' denote any corresponding values of x and y ; 
then ?/' = ax' + &. (2) 

When x = x' -\- Ax, y = y' + Ay ; hence, from (1) we have 

?/ + A7/ = a.(x' + Ax) + 6. (3) 

Subtract (2) from (3) ; then, as x' is any value of x, we have in general 

Ay = aAx, or A (ax + b)= aAx. (4) 

2. The ratio of A (x^) to Ax is the variable 2 x + Ax. 

When X = x', we find by the method above 

A (X-) = (2 X' + Ax) A,r. 
Hence, as x' is any value of x, we have in general 

A (x--^) = (2 X + Ax) Ax. 

3. Trove that A/(x) =/(x + Ax) -/(.r) for any vahu- oi x. 

4. Find A (ax- + ex); A(.r); A{ax-^ + hx)\ A((\r«); A((\r^-^r-). 

When ./•(•'•) ^ ''•*•- + <■•'•, /(•'• + -^•'■) ^" 'f (•'• + -^-i")'- + c{,x + A.r); 
.-. A ((IX- + ex) = a (.(• f Ax)- 1" (• (.(• I Ax) - {ax- -f ex) 
=; (2 ((X + (rAx -|- (•) Ax. 



6 DIFFERENTIAL CALCULUS. 

9. Uniform and non-uniform change. When the incre- 
ments of one of two variables are i:)roportional to the corre- 
sponding increments of tlie other, either variable is said to 
change uniformly with respect to the other. When the corre- 
sponding increments of two variables are not loroportional^ 
either variable is said to change non-uniformly with respect 
to the other. E.g. the rectangle ABCD in § 10 changes 
uniformly with respect to its variable base AB, while the 
triangle ABC changes non-uniformly with respect to its base. 

Hence, f{^) changes uniformly or nioyi-uniformly with 
respect to x according as the ratio of A/(.x) to ^x is con- 
stant or variable. 

Ex. 1. With respect to cc, does ax change uniformly or non-uniformly ? 
ax -h 6 ? a;2 ? a;^ ? ax- + hx'^ ax^ -c? 

f(x) changes uniformly luith respect to x ivhen, and only 
ivhen, f(x) is a linear function of :^. 

For if f{x) = ax -{- h, A/(cc)/Ax = a ; hence, f{x) changes 
uniformly with respect to x. 

li f(x) is not linear in x, Af(x)/Ax is variable; hence, 
f(x) changes non-uniformly with respect to x. 

Ex. 2. Show that y changes uniformly or non-uniformly with respect 
to x according as the point (x, y) traces a straight or a curved line, 

10. Measure of rate. The rate of a variable is the rapid- 
ity of its change. Different variables have different rates, or 
change at different degrees of rapidity ; and in general the same 
variable passes through different values at different rates. 

To measure the rates of variables, we fix upon some unit ; 
that is, we select the rate of some variable as a iniit of rates. 
For convenience, the rate of some increasing variable is always 
chosen as a unit of rates. If the rate of x is assumed as the 
unit of rates, we have the following definitions : 

I. If a variable y changes uniformly with respect to x, the 
measure of the rate of y is the increment of y corresponding 
to the increment 1 of x. 



MEASURE OF RATE. 



By § 9, ax + 6 changes uniformly with respect to x. 

If Ax = 1, A (ax + &) = aAx = a. 

Hence, ax + b changes uniformly at the rate of a to 1 of x 

Thus, 4 X + 7 changes uni- 



formly at the rate of 4 to 1 of 
X ; and — 8 x + 9 changes uni- 
formly at the rate of — 8 to 1 
of X ; that is, — 8 x + 9 decreases 
at the rate of 8 to 1 of x. 

Again, if AD is constant, and 
AB is increasing, and i>ilf equals 



D 



C rj 



B e 



h N 






a unit of the base, the rectangle ABCD increases uniformly at the 
BMNC to a unit of the base. 



rate of 



II. If a variable changes non-uniformly with respect to x, 
the measure of its rate is what its increment corresponding to 
the increment 1 oi x would he if at the value considered its 
change became uniform with respect to a?. 

Conceive a variable right triangle with the constant angle A as gen- 
erated by the perpendicular moving to 

the right. The rate of the area of the ^ ^' O 

triangle at the value ABC is evidently 
equal to the rate of the area of the 
rectangle BCOII. Hence, if BII equals 
a unit of the base, the rate of the area 
of the triangle at the value ABC is 
BIJOC to a unit of the base. Now, 
evidently, BIIOC is what would be the 
increment of the area corresponding 
to the increment BII, if at the value 
ABC the change of the area became 
uniform with respect to the base. 

Suppose BC = 2AB, and let x and 2x denote Iho nninbor (> 
the base and perpendicular, respectively ; 

then area, A JiC = x- units, 

and area /)7/()(^ - -J x units, siiico 7.7/ = 1. 

llenc(% X- chang(\s at tlie rate of '2 .r to 1 oi x. 

When X = 2, \a\ wIumi x passes tlirondi tlii' value 2. x- chanuvs at ihe 




rate of 4 to 1 of 



WlUMl X 



X- rhaiim's at the rati' of 1(? to 1 oi x 



8 DIFFERENTIAL CALCULUS. 

When the rate of x is the unit of rates, the measure of the 
rate of a variable is called its x-rate. 

The rate of a variable will be positive or negative according 
as the variable is increasing or decreasing ; and conversely. 

11. Differentials. The dijferentials of variables which 
change uniformly with respect to the same variable are their 
corresponding increments. 

The differentials of variables which change non-uniformly 
are what tvould he their corresponding increments if at the 
corresponding values considered the change of each became 
and continued uniform with respect to the same variable. 
Hence the differential of a variable will be positive or negative 
according as the variable is increasing or decreasing. 

The differential of a variable is denoted by writing the 
letter d before it ; thus, dx read '^ differential cc" is the symbol 
for the differential of x. When the symbol of a function 
is not a single letter parentheses are used ; thus, d (x^) and 
dix^ — 2 x) denote the differentials of x^ and x^ — 2x, respec- 
tively. 

In the first figure of § 10, 

\i Be = d (base), BegC = d (rectangle) ; 

if Bf=d (base), BfhC = d (rectangle). 

In the second figure of § 10, 

if Be = d(base), BegC = d (triangle); 

it Bf = d (base) = dx, BfhC = d (triangle) = d (x^). 

... ^ = ^^^ = BC = 2x = the x-rate of x^ § 10 

dx Bf JO 

which is a particular case of the next theorem. 

12. dy/dx, a rate or a ratio of rates. 

I. When the rate of x is the unit of rates. 
In this case dx is positive. Let n be so chosen that ndx 
will be equal to 1 ; then nd// will denote what would be 



RATIO OF RATES. 9 

the increment of y corresponding to the increment 1 of x, if 
at the vahie considered the change of y became uniform with 
respect to x ; hence, 

ndy = the a:-rate of y. § 10 

.'.—- = — 7^ = ndy = the a^-rate of y. 
ax ndx 

That is, tJie dijfevential of one variable divided by that of 
another whose rate is the unit of rates is equal to the rate 
of the first variable. 

II. When the rate of x is not the unit of rates. 
Let the rate of v be the unit of rates ; then 

dy dy / do the v-rate of ?/ 

dx dx/dv the v-rate of X 

That is, the ratio of the differentials of any two variables is 
equal to the ratio of their rates. 

In this case dx may be either positive or negative. 

CoK. 1. If y =f(x), y is an increasing or a decreasing func- 
tion of X according as dy j dx is positive or negative. (AVhy?) 

Ill practical affairs and physical science the more common unit of rates 
is the rate of time. Thus, we speak of a distance as increasing at the 
rate of 20 miles an hour, meaning thereby that if at the value considered 
the change of this distance became uniform with respect to time, its incre- 
ment in an liour would he 20 miles. 

CoK. 2. If t denotes the number of units in a portion of 
time, then dy/dt, or the z^-rate of y, gives the time-rate of //. 

13. Tlie meanings of A//, dy, and the .r-rate of // should 
be carefully considered. A// denotes tlie actual increment id" 
y corresponding to Aa*. dy denotes what tlie increnuMit of // 
corresponding to the increment dx would b,\ if at the value 
considered the change of // becanu^ nnifonn with iw^pect to .v. 

Hence, dy = Ay only when //changes uniforndy with respci-t 
to X and dx = Aa*. 



10 DIFFERENTIAL CALCULUS. 

The ic-rate of y, which equals cly / dx, is what the increment 
of y corresponding to the increment 1 of a^ would he if at the 
value considered the change of y became uniform with respect 
to X. When the rate of x is not the unit of rates dy /dx 
means simply the ratio of the rate of y to that of x. 

14. The problem of the Differential Calculus is to measure 
and compare the rates of change of continuous variables when 
the relation of the variables is known or given. 

The inverse problem of finding the relation of the variables 
themselves when their relative rates are known is the prohlei)i 
of the Integral Calculus. 

EXAMPLES. 

1. With respect to u, does av change uniformly or non-uniformly ? 
au + c ? v2 p av'^2 v'^ + av'^ sin v ? 

2. What is the i-rate of ai ? ai + 6 ? f-? t^ + at? t^ + bt + c? 

3. What is the u-rate of au ? av + c? v-? v^ + av -\- c? 

4. Conceiving a square as generated by two of its sides, illustrate that 
d (x^) = 2 xdx, and that x- changes at the rate of 2 x to 1 of x. 

5. Conceiving a cube as generated by three of its faces, illustrate that 
d (x^) = 3 x^ dx, and that x^ changes at the rate of 3 x^ to 1 of x. 

6. Conceiving the point (x, y) as tracing any curve, draw the lines 
which represent dij and dx at any point P, and show that dy / dx is the 
slope of the tangent to the curve at P. (See § 33.) 



7. For wh at real value s of x is Va^ — x^ real and continuous ? 
'x2 - a2 ? V(a + 6)2 - x--^ ? a/x? a/{c-x)? cotx? logx? 

8. If /(x, ?/) = e^ — e— 2/, prove 

/(3x, S7j) = [f{x, y)Y + 3e-e-2//(x, y). 

9. If /(.)^^-^, show that f(')-f(-) ^1^. 

e + 1 i+f{d)-f{x) i + to 



CHAPTEE, II. 
DIFFERENTIATION. APPLICATIONS. 

15. Differentiation is the operation of finding the differen- 
tial of a function. The sign of differentiation is the letter d ; 
thus d in the expression d{x''') indicates the operation of dif- 
ferentiating x^, while the whole expression dioi?) denotes the 
differential of x^. 

In the folloiuing formulas, u, y, v, w, and z denote different 
functions of x. The rate of x will be used as the unit of 
rates. 

16. If u = y, du = dy. [1] 

That is, the differentials of equals are equal. 

For if u continually equals y, it is self-evident that ii and // 
must change at equal rates ; that is, 

dit J dx = dy / dx ; .' . du = dy. 

For example, if y^ = 4 ax, d (y^) = d (4 ax). 

17. d(a)-o. [2] 

Tliat is, f//e differe^itlal of a co/istant is rjcro. 
For the rate of any constant <( is zen) ; tliat is, 
da/ dx = ', .'. da — 0. 

18. d(u + y H h z + a^ du -f dy ^ + dz. [r>] 

'^rhat is, f/ic diff'roidal of (f /)o/////o)n /<rl is tlir stun of the 
differentials of its terms. 



12 DIFFERENTIAL CALCULUS. 

Por it is self-evident that the rate of the sum u -\- y -\- ' • ' 
-\- z -{- a is equal to the sum of the rates of its parts u, y, • - ; z, 
and a ; that is, 

d(7i -\- y -\- ■ ' ' -\- z + a) _ du _,dy dz da 

dx ~ dx dx dx dx 

Multiplying by dx, since da = 0, we obtain [3]. 

For example, d (o x-~-^x^ — 2) = d (3 x^) + d (— 4 a:-) + d (— 2). 

19. d(au) = adu. [4] 

That is, tJw differential of the product of a constant and a 
variable is the product of the constant and the differential of the 
variahle. 

By § 9, au changes uniformly with resjoect to u. 

By § 8, t^{ciiL) = a\u. 

• Hence, by the definition in § 11, we obtain [4], 

For example, d (o (ufi) — ?ia-d (x-3), and di-\ = d(~ • zj = - dz. 

20. d (logaU) = m • du/u, where u is positive. [5] 

That is, the differential of the logarithm of a variable is the 
modulus of the system into the differential of the variable divided 
by the variable. 

For, n being a general constant, let 

u = ny. (1) 

• • • loga^^ = log„7i + log jy. (2) 

From (2), by [1], [2], and [3] we obtain 

dQo^jt) = d(\og,y). (3) 

Differentiating (1) and dividing by (1), we obtain 

du/u = dy /y. (4) 



LOGARITHMIC FUNCTIONS. 13 

Dividing (3) by (4), we obtain 

d (log„i^) '.du/u = d (log„7/) : dy / y. (5) 

It remains to prove that the equal ratios in (5) are constant. 
Let m denote the common ratio in (5) when 2/ — 2/' 5 then 

d (log^-w) = m • du I u, (6) 

when u = ny'. But, as tz is a general constant, 7iy' denotes any 
number; hence (6), or [5], holds true for all values of u, m 
being a constant. 

The constant m is called the modulus of the system of 
logarithms whose base is a. 

The modulus of the common system of logarithms, obtained 
in § 97, is 0.434294 • • •. 

From the nature of logarithms we know that log^w changes the faster 
the smaller we take a. From [5] we learn that logaU changes at the rate 
of m/u to 1 of u. Hence, the modulus vi varies with the base a. 

Ex. The number u changes how many times as fast as logiow when 
u = 2560 ? 

du u 2560 ^_-^ 

■Jn T = ~ — TTT^VT^. — 5895, nearly. 

That is, u changes nearly 5895 times as fast as its common logarithm 
when u = 2560. 

21. Natural logarithms. The system of logarithms wliose 
modulus is unity is called the Naperian or nctfural system. 
The symbol for tlie base of the natural system is e. 

Hence, d(log,.u) r: du/u. [(>] 

Natural logarithms are evidently the simplest and nn^st 
natural for analytic purposes. 

Hereafter when no base is writttMi, e is uiuhM-stood. 

In the naiuval syslcm (In n •</ (lnu' //); thai is, [\\c number u I'liauiios 
n times as fast as ils natural lo^arilhni. 



14 DIFFERENTIAL CALCULUS. 

22. d(uy)=ydu + udy. [7] 
When u and y are botli positive, we have 

log {uij) =log«H-log y; 

uy u y ^ L J 

.'. d (uy') = ydu + udy. (1) 

Multiplying (1) by ah, by [4] we obtain 

d (au ■ by) =^ by • d (aii) + au • d(by). (2) 

Since a and ^ are general constants, au and by denote any 
variables, real or imaginary. Hence [7] holds true for any 
real or imaginary values of u and y. 

23. d(uvyz- • •) 

= (vyz • • •) du + (uyz • • •) dv + (uvz • • •) dy H . [8] 

That is, the differential of the product of any number of 
variables is the sum of the p>^'oducts of the differential of each 
into all the rest. 

If in [7] we put viv for u, we obtain 
d (yivy) = yd (yiu) H- vwdy 

= wydv + vydiv + vwdy. (1) 

By repeating this process the theorem is proved for any 
number of variables. 

If i; = 10 = ?/, (1) becomes diy'^) = 3 y^dy. 

24. d(u/y) = (ydu-udy)/y2. [9] 

That is, the differential of a fraction is the deiiominator into 
the differential of the numerator minus the numerator iiito the 
differential of the denominator, divided by the square of the 
denominator. 



EXPONENTIAL FUNCTIONS. 15 

Let z ^= u I y \ then zy = u. 

.'. ydz + zdy = du ; by [1], [7] 

.\dz = {du-zdy)/ij, 



<i) 



die — (u/y)dy ydu — udy 



or ^. , 2 

y y 



Cor. d(a/y) = -ady/yl [10] 

-^ ^a 1/dci — adii adit 

For d- = — — = r^> since da = 0. 

y y y 



25. Differentials of u^, b^, u". When u is positive, 

d(u^') die 1 rtt-i 

.-. m ^^ = m7/ — + log„i^ . dy ; by [5] 

.•.d(uy)=yuy-idu + uy log,,u dy/m. [11] 

Putting Z» for u in [11], by [2] we obtain 

d(by) = by log^^b dy/m. [12] 

Putting 71 for y in [11], by [2] we obtain 

d(u") = nu"-^du. [i;>] 

Multiplying [13] by c", by [4] we obtain 

d(<'i(y = v(<'u)"-'d(('u). (1) 

Since cx^ in (1) denotes ;iny variable base, [l'>] ImKls true 
for a,ny value ol" /^ ])ositive or nei;a(iv(\ real or imaginary. 
The function //■' is continuous (Uily wluui // is |u\^itive : hence 
[11] and [12] are liniitt^l to positive vabuvs id" // and />. 



16 DIFFERENTIAL CALCULUS. 

26. Stating [13] in words, we have 

The differential of a variable base affected ivith a constant 
exponent is the j^^oduct of the exponent, the base with its exponent 
diminished by one, and the differential of the base. 

Cor. dVu = du/2Vu. [14] 

For diii^'"") =iu-^'^du = du/2-\^u. 

27. Stating [12] in words, we have 

The differential of a7i exjoonential function luith a positive 
constant base is the function itself into the logarithm of the base 
into the differe7itial of the exponent, divided by the modulus of 
the system of logarithms used. 

Cor. In the natural system, m = 1 ; hence, as log e = 1, 
from [12] we have 

d(b5^)=bnogbdy; [15] 

and d (e^) = e^ dy. ^ [16] 

28. Comparing [13] and [12] with [11], we see that 

The differential of an exponential function with a p)ositive 
variable base can be obtained by first differentiating as though 
the exponent luere constant, and then as though the base were 
constant. 

EXAMPLES. 

By one or more of the preceding formulas exclusive of [5], [6], [11], 
[12], [15], [16], differentiate 

1. ?/ = x3 - 8 X + 2 x2. 

d?/ = d (x3 - 8x + 2 x2) by [1] 

= d (x3) + d (- 8 X) + d (2 x2) by [3] 

= 3 x2d!x - 8 dx + 4 xdx. by [4], [13] 

.-. d?/ /dx = 3 x^ — 8 + 4 X = the x-rate of y. § 12 



ALGEBRAIC FUNCTIONS. 17 

That is, ?/, or x^ — 8 X 4- 2 x^, changes at the rate of 3 x^ — 8 + 4 x 
to 1 of X ; or y changes 3 x^ — 8 + 4 x times as fast as x. 

When X = — 4, 2/ is increasing at the rate of 24 to 1 of x ; 
when X = 0, ?/ is decreasing at tlie rate of 8 to 1 of x ; 
when X = 5, ?/ is increasing at the rate of 87 to 1 of x. 

2. ?/ = 3 ax2 — 5 )2X — 8 ?n. dy = {Qax — b n) dx. 

Note. At first the student should give the meaning of each differ- 
ential equation which he obtains. 

3. y = 6 ax^ — 3 b'^x^ — ahx^. dy = (10 ax — 9 b'^x'^ — 4 abx^) dx. 

4. y = a^ + 5 62x3 + 7 a^x^. dy = {15 ¥-x'^ + 35 a^x^) dx. 

5. ?/ = ax3/2 + 6xi/^ + c. dy/dx = {Sax + b)/2V^. 

6. ?/ = (& + ci^'^)^ '^' dy = ^{b + 0x2)1 /4 axdx. 

1. y = {ex + 6x3)4 / 3. dy = ^{cx-\- 6x^)1 / 3 (c + 3 6x2) dx. 

8. 2/ = (1 + 2x2) (1 +4x3). (Z?/ = 4x(l +3x + 10x3)tZx. 

dy = {l +2x2)(Z(l +4x3) + (1 +4x3)cZ(l +2x2) 

9. y = {x + lf{2x-\f. d?/ = (16 x + l)(x+ 1)4(2 X- 1)2 tZx. 

1/% / . V / ■ ciy a — ox 

10. ?/ = (a + x) Va — X. — = — -j== • 

dx 2 V a — X 

11. ?/ = (X2 + 1)2 (2 X2 + X)3. 

dy = (x2 + 1) (2 x2 + x)2 (20 x3 + 7 x2 + 12 X + 3) dx. 

12. ?/ = (1 - 3 x2 + G x4) (1 + x2)3. dy = 60 x^ (1 + x2)2 dx. 

^ 13. y = x^{a + 2 x)3 (a — 3 x)2, 

dy = G x^ {a + 2 x)2 {a — 3 x) (a2 — ax — 1 1 .r'-) dx. 

14 — ^ + ct^ (^U _ ^ — g- 

' ^ ~ X + 6 ' dx ~ (X + 6)'-i * 

_ (X + 6) d {X + gS) - (x + a2) g (.p + ^) 

'^^ ~ (X + 6)2 

IK — ^^'"^ . <■]£ _ 8 a-.v' ' — i X'"' ^ 

16. ?/ = Vrtx- + 6x + c. - - = — , = • 

<"5 2Vrt.r- + 6.r + c 



18 



DIFFERENTIAL CALCULUS. 



17. y 

18. y 

19. y 

20. y 

21. y 

22. y 

23. y 

24. ?/ 

25. y 

26. 2/ 

27. ?/ 



2 a;2 - 3 

4x + x2' 



d?/_8x2 + 6x + 12 



n/ 



1 +X 

1 -x' 
x^ 



(1 + X)2 

2x2-1 



xVl +X2 

a2-52 
(2ax — x2)3/2 

Vax + Vc2^. 
x« + 1 



x«-l 


6x 


V2 ax — x2 


x« 


(1 + x)« 


1 



dx 


(4 X + X2)2 


dy_ 


1 


dx 


(1-X)V1-X2 


dy_ 


3 X2 + X3 


dx 


(1 + X)3 


dy_ 


1 +4x2 


dx 


x2(l+x2)3/2 


dy_ 


3 (a2 - &2) (X - a) 


dx 


(2ax — x2)5/2 


dx 


V^ + 3cx 
2Vx 


dy_ 
dx 


2nx'*-i 
(x« - 1)2 


dy _ 


abx 


dx 


(2ax — x2)3/2 


dy _ 


?ix"-i 



{a + x)"* {b + x)» 

{a + x)»* (& + x)«. 

d?/ = {7W (& 4- x) + 



28. 


^ (1 + X2)« 


29. 


?/ = xi2 (6-3 x)4 (c + 4 x)3. 




dy = 12 xii (6 — 3 x)3 (c 


30 


1-x 

2/ = 




Vl + X2 


31. 


Vx2 - a2 

y =^ — I 



dx (l+x)«+i 

dy _ _ m{h + x) + n{a + x) 
dx~ (a + x)''» + i(6 + x)«+i 



w(a + x)}(a + x)"'-i(6 + x)«-idx. 
dx~ {1 + x2)« + i* 

+ 4 x)2 (6c — 4 ex + 5 6x — 19 x2) dx. 
dy ^ - (1 + X) . ■ 

(ZX (l+x2)3/2 

t^?/ _ a2 
dx x2V'x2 — a2 



LOGARITHMIC FUNCTIONS. 19 



32,. = ^ ^Jl = ^f^±l^ + 2A 

Va2 -\- x'^ —X dx a?- ^ ^ a'^ + cc'^ J 

Rationalize the denominator before differentiating. 

-^ = — , 4 x^. 

dx Vx2 + 1 



33. 


^ = 


X + Vl + x'^ 


34. 


2/ = 


X 


Vx2 + a2 _ (^ 



dy _ _a^ 
dx X? 



(1 + ^=^=1 



EXAMPLES. 
By one or more of the sixteen preceding formulas, differentiate 

-, 1 / o I ^ dy 2x^r\ 

1. j, = log(x- + x). Tz = W+^- 

2. y = logaX^ = 3 log,,x. 

3. y = log„Vr^ = ^I„g„(l - x»). I = -|^. 

4. ?/ = X log X. dy = (log X + 1) dx. 

5. 2/ = log ^ = log(l + Vx) — log(l — Vx). 

1 — Vx 

> 8. y = log (Vl + X2 + Vl - X2). ^-^rrl/'-^lL::::^ -l-lV 

dc X Wl - x-t / 

9. 7/ = (l0gX)3. 10. ?/ = (• lo^'"'-. 

11. ?/ = x-^. dy = X-'' (log X + 1) dx. 

12. y = x-^^. |^( = x'-'>'/ log .r (U^g .r + + ^ ) . 

Here log ?/ = x-^' log x ; .-. — = x-'" — + lou" x I .r'' (lou; x + 1 (f.r. 
' y x - I \ ^ 

13. V = X' . — — X' e' 

dx X 



20 DIFFERENTIAL CALCULUS. 



14. 


y = e^. 


dy = e^x^{\ogx + l)dx. 


15. 


y = xio?^. 


dy = 2x^°s^-i logx-c?x. 


16. 




dy _2a'' log a ^ 
dx {a^ + 1)2 


17 


X 


dy _ 1 




^ ''^Vi+x^' 


dx x{l + x2) 


18. 


y = log (log x). 


dy _ 1 

dx x log X 


19. 


y^x^' 


i=a)>-:-o- 



20. y = J-k^±-. ^JL= ^-2^-2 . 

^x^ + x + l dx 2V1 -X^X2 + X + 1)3/2 

In example 20 and some that follow, pass to logarithms. 



21. ?/= """ 



(a + xY 

^~(a2-x2)l/2 r/r -^^/«2_ ^.2^3/21^ ^^) • 

pX 

23, J/ = log^^pj,- 

24- ^ = ^''^- ax 

25. 2/ = e^(l— x3). d?/ = e^ (1 — 3 x2 — x3) dx. 

26. ^^ ^^-^^ #_ 4 . 



d?/_ 


nax"-i 


dx 


(a + x)« + i 


dy_ 


_, 2 a2 - x2 


dx 


"""(a2-x2)3/2' 


d2/_ 


1 


dx 


1 +e- 


d?/_ 


(l-logx)xi/^ 



ga: + e-x ax (e^ + e-^)2 

28. 2/ = (a^ + 1)2. dy = 2a^ {a^ + 1) log a dx. 

^ 29. y = a'^. 30, 2/ = — ^• 



DERIVATIVES. 21 

29. Derivatives. The ratio of tlie differential of a function 
of a single variable to that of the variable is called the deriva- 
tive of the function. Thus, the derivative of 7/ as a function 
of X, i.e. dy / dx, is the i^-rate of y or a ratio of rates. 

The derivative of /(x) is denoted by/'(cc). 

That is, df(x) /dx= f'(x). 



The operation of finding the derivative of a function of x is 

noted by -^. Thus ^ (3 x') = "^-^^ = 15 x\ 
dx dx^ ^ dx 

A derivative is often called a differential coefficient. 



EXAMPLES. 

In each of the following equations find the derivative of the implicit 
function y : 

1. x^ + y^ = 3 axy + c. 

d (x3 + ?/) = d{S axy + c) ; by [1] 

.-. 3 xMx + 3 y'^dy = 3 aydx + 3 axdy ; 
dy _ ay — x'^ 
■ ' (Zx y'^ — ax 



dy _ 4xy -hh 
dx 3 y-^ — 2 X- 



2. y^ = 2 x-v/ + 6x. 

3. ?/'^ = 4px. 4. a'"?/- + 6-x'- = (1-62. 



5. x3 + 3 ax?/ = — y^. 



dy _ _ X" + (ty 
dx y'^ + ax 



Q •^A.!lZ=i '-111= _ ('^\ ( -\ 



x^ _^ r 

a"' b'" " dx 



dy _ y {I — x ) ^ 
dx ~~x(?/ — 1) 



7. e^ + !i — xy. 

Passing to logarithms, wo \\\\\c x + // = U\u- x + log //. 

dx .c"'^ — .ry log .r \x ^ logy— 1 



22 DIFFERENTIAL CALCULUS. 

9 ^, + ,^ ,_, ^^ xy\og{xy) + y{y + x) 

dx xy log {y/x)-\-x{y — x) 



10. {y + xy = x^ay. 



dy _ y + {x + y) [log x — log (x + t/)] 
dx • X — {x + y) log a 



30. The velocity of a moving body is the time-rate of the 
distance traversed by it. Let s denote the distance, t the 
time, and v the velocity; then v = the time-rate of 5 = ds/dt. 



EXAMPLES. 

1. The area of a circular plate of metal is expanding by lieat. When 
the radius passes through the value 2 in. it is increasing at the rate 
of 0.01 in. a second ; how fast is the area increasing ? 

Let X = the number of in. in the radius, 
and y = the number of sq. in. in the area. 
Then y = irx'^ ; .-. dy/dt = 2 ttx • dx/dt. (1) 

When X = 2, dx/dt = 0.01; .-. dy/dt = 0.04: ir. 

That is, the area is increasing at the rate of 0.04 tt sq. in. a second 
when it passes through the value considered. 

2. When the radius of a spherical soap-bubble equals 3 in. it is in- 
creasing at the rate of 2 in. a second ; how fast is its volume increasing ? 

Ans. 72 IT cu. in. a second. 

3. A boy is running on a horizontal plane in a straight line towards 
the base of a tower 50 metres in height. How fast is he approaching the 
top when he is 500 metres from the foot, and is running at the rate of 
200 metres a minute ? ^^s. I99 metres a minute. 

4. A light is 4 metres above and directly over a straight horizontal 
sidewalk, on which a man If metres in height is walking away from the 
light at the rate of 50 metres a minute. How fast is his shadow changing 

in length ? 

Let AE he the sidewalk, B the position 
of the light, and CD one position of the man. 

Let y = the number of metres in CE, 
and X = the number in ^ O ; then 
2/ + X :?/ = 4 : 5/3, etc. 




RATES OF CHANGE. 23 

5. In problem 4 sliow that tlie farthest point of the man's shadow is 
moving at the rate of 85 f metres a minute. 

6. Tlie altitude of a variable cylinder is constantly equal to the 
diameter of the base. When the altitude equals 6 metres it is increasing 
at the rate of 2 metres an hour ; how fast is its volume increasing ? How 
fast its entire surface ? 

Ans. 54 IT kilolitres an hour ; 36 tt centiares an hour. 

7. Find the length of a side of an equilateral triangle when its area is 
increasing in sq. in. 30 times as fast as a side is in lin. in. 

Let X = the number of lin. in. in a side of the triangle, 
and y = the number of sq. in. in its area ; 
then y = ^Vsx'^ ; .-. dy = ^Vsxdx. 

Hence, y changes ^V3 a: times as fast as x; therefore, when y 
changes 30 times as fast as x, i Vs x = 30, or x = 20 V3. 

8. On the parabola ^/^ = 8 x, find the point at which y changes at the 
rate of 2 to 1 of x. ^^. (1 /2, 2). 

9. On the ellipse x^/a'^ + y'^/b^ = 1, find the points at which y changes 
at the rate of c to 1 of x. 

Ans. (if ca'^/^b-2 + a^c% ± h'^/^b- + aV^). 

10. One ship was sailing south at the rate of 6 miles an hour ; another 
east at the rate of 8 miles an hour. At 4 p.m. the second crosses the 
track of the first at a point where the first was two hours before. "When 
was the distance between the ships not changing ? How was it changing 
at 3 P.M. ? at 5 p.m. ? 

Let ^ = the number of hours in the time reckoned from 4 p.m., 
time after 4 p.m. being +, and time before — . Then 8 t miles and 
(6^ + 12) miles will be respectively the distances of the two ships 
from the point of intersection of their paths, distances south and 
east being +, and distances west and north being — . 

Let y = the number of miles between the shij^s ; (hen 

2/2 = (8 0- + (6 i + 12)2. 

di/ 100« +J2 

■'■ dt ~ [(54 r- +{{)t + 12V-^]» /•-' ' 
Wlion di//dt - 0, 100 / -f 72 = 0, or / == - 0.72. 
Hence, the distance between the ships was uo\ i-hanging at 4.'>.2 
minutes before 4 p.m., or at 1().8 minutes after ;'> v.m. 

Ans. Diminishing at (he rate of 2.8 iniUvs an hour ; inereasing at 
tlie rate of 8.7;> niilos an hour. 



24 DIFFERENTIAL CALCULUS. 

11. Two straight lines of railway intersect at an angle of 120°; on one 
line a train is approaching the intersection at the rate of 30 miles an hour, 
and on the other a train is receding from it at the rate of 40 miles an 
hour. Wlien the first is 10 miles from the intersection, the second is 
20 miles from it ; find the rate of change of the distance between the 
trains at that instant. 

Let z = the number of miles between the trains, and x and y 
respectively the numbers of miles between them and the intersection 
of the tracks ; then the rates of x and y respectively will be — 30 
and 40 an hour, and 



z2 = x^ + y^ + xy. 
dz_l_{,^, , dx 
dt ~ 2z 



1 r ,^ , .dx..^ . .dy\ 40 /- 



Hence, at the instant considered the trains are separating at the 
rate of -Y- "^7 miles an hour. 

31. Tangent. If a secant RS be moved or revolved so that 
two of its points of intersection with a curve coincide at some 
point P, the line BS in this position is the tangent to the 
curve at P. 

32. Slope. "* The tangent to a curve at any point F has 
the direction of the curve at that point. 

Let ^ denote the angle XBP (§ 33, fig.) ; then, when P 
moves along the curve, <^ varies and gives the direction of the 
curve at P relative to the x-axis. 

Tan <^ is called the slope of the curve at P. 

33. Geometric meaning of dy/dx. Let mPP^ be the locus 
of y =f(x). Let s- denote the length of the path traced by 
the point {x, y). 

* This statement, if not sufficiently evident, may be proved as follows : 
When the direction of the arc PaP' (§ 60, fig.) changes continuously 
(this arc can always be made so small that its direction will change con- 
tinuously), the secant PP' has the direction of the arc PaP' at some point, 
as a. When P' is made to coincide with P, a also will coincide with 
P ; hence, the tangent at P has the direction of the curve at P. 



EQUATION OF TANGENT. 



25 




N X 



Starting at m, suppose (x, y) to move along the curve to P 
and thence along the tangent 
PA. Then, when x = OM, 
the change of x and that of 
y/ will both become uniform 
with respect to s. 

Hence, PA, PE, and EA 
will represent ds, dx, and dy, 
respectively, when x = OM. 

Hence, di//dx = tan ^. (1) 

That is, dy/dx, orf(x), equals the slope ofy =f(x') at (x, y). 

Since the relative rates of y and x determine the direction of motion 
of the point (cc, ?/), equation (1) follows directly from § 12. 

CoK. 1. From the right angled triangle PEA we have 

els'" = dx" + dy'', (2) 

dy I ds = sin ^, dx / ds = cos ^. (3) 

CoR. 2. If PA represents the velocity at P of the point 
(x, y) in the direction of its path, PE and EA will represent 
the component velocities at P of the point (x, y) in the direc- 
tion of the axes. 

The figure given above illustrates the difference between an incroiut.'nt 
and a differential. 

Eor example, if Ax = PE = dx, A,y = EP' = dij + A P'. 

If the curve were concave downward at P, A?/ would be less than (///. 
A?/ and dy are equal when and only when the locus is a straight line. 

34. Rectangular equation of a tangent. Lot y = f{^') bo 
the rectangular 0{|uation of any piano curvo. and lot dy' /dx' 
denote the value of dy / dx for the point (.r', y'^ ; tlion tbo 
equation of the tangiMit to tbo curvo y - t\-A at tlio point 
(x\ y') is 



y 



dy 
dx 



,(■>■ -•'■'). 



(=>) 



26 



DIFFERENTIAL CALCULUS. 



For line (a) evidently passes tlirougli {x\ y'), and by § 33 it 
has the slope of the curve at that point. 

Ex. Find the equation of the tangent to the parabola y- = \]px. 

Here dy /dx = 2p/?/ ; .-. dy'/dx' = 2p/y'. 

Substituting in (a), we obtain as the required equation 

2r> 
y-y^ = y{x- X'). 

Since y'^ = 4:px% equation (1) becomes by reduction 
yy' = 2p{x + x'). 

CoK. Intercept of tangent on a:-axis = x' — y'dx' /dy'. 
Intercept of tangent on ?/-axis = y' — x'dy' /dx'. 



(1) 

(2) 

(1) 
(2) 



35. Rectangular equation of a normal. The normal at 
the point (x', y') passes through (x', ?/') and is perpendicular 
to the tangent at that point ; hence, its equation is 



dx' 



(b) 



For example, the equation of the normal to the parabola y^ = ipx at 
the point {x', y') is 

y-y' = - {y' /2p) (x - x'). 



36. Subtangent, subnormal, tangent, normal. Let PT 

be the tangent at the point F{x', y'), and PS the normal. 




Draw the ordinate PM-^ then TM is called the siihtangent, 
and MS the subnormal. Hence, 



TANGENTS AND NORMALS. 27 

Subt. = TM=MP cot (/> = ijdx' ldy\ (1) 

Subn. = MS = MP tan c^ = y'dy'/dx'. (2) 



Tan. = TF =\J MF' -f TM^ = v/' Vl + (dx'/dyf. (3) 
Norm. = SF =\/mf' + JlS"" = ij^l -^ (cly' / dx'f. (4) 

If the subtangent is reckoned from the point T, and the 
subnormal from M, each will be positive or negative accord- 
ing as it extends to the right or to the left. 

Note. The problem of tangents was foremost among the problems 
which led to the invention of the Differential Calculus. 



EXAMPLES. 
The equations of the tangent and the normal to 

1. The circle, x- + if-' — r^, are yy' + xx' — ?-2, and x'y — y'x. 

2. The ellipse, x^/a^ + y-/62 = 1, are xx' /or + yif /h- = 1, and 

y —y' = {o?'ij'/lP'X') (x — x'). 

3. The hyperbola, x^/a^ — y"^ /If- — 1, are xx' / or — yy' /h- = 1, and 

y — y'= — {a'^y'/h'^x') {x — x'). 

4. The cissoid, y2 = __-, are y-y=± ^^_^y,,, {^ " ^ ), 

(2 a — x')^/2 

and y — y' — =f —. ■ — (x — x'). § 155, fit?. 3. 

Vx' (3 a — x') 

5. Tlie hyperbola, xy = /;^, are x'y + y'x = 2?//, and 

y'y — x'x = y"- — x"^. 

6. The circle, x"' + y- = 2 rx, are ?/ — //' = (x — x') (/ — x')///', and 

// — //' = (•'• — ^I'l!/'/ (•'■' — '•)• 

7. 1^'iiid llio tM|n;iti(Uis of tlu^ tangt-nt and {he normal to tlu' i'ur\o 
?/'-^ = 2x- — x"\ (1) at tlu> point, whosr abscissa is 1, (2) at tlic point whoso 
abscissa is — 2. 

8. Find {ho slopi^ ()\' tlu> cnrvc //- = X'' + 2x* at x -' 2. 

9. Find th(> sloi)t' of tlu> i-nrvo // = x-' — //- + 1 at x -- 2 ; x = 1 ; x = 0; 
a:= -L 



2S DIFFERENTIAL CALCULUS. 

10. At what point on y'^ = 2 r^ is the slope 3 ? At what point is the 
curve parallel to the x-axis ? ^,^5 /2 4) • (0 0). 

11. At what angle does y- = 8x intersect 4 x^ + 2 ?/2 = 48 ? 

The points of intersection are (2, 4) and (2, — 4) ; the slopes of the 
curves are 4/?/ and —2x/y respectively, which for the point (2, 4) 
become 1 and — 1. Hence, the curves intersect at right angles at 
(2, 4). 

12. At what angles does the line Sy — 2x — 8 = cut the parahola 
y^ = Sx? Ans. tan- 10.2; tan- 10.125. 

13. The cissoid y^ = x^/ {2 a — x) cuts its circle x- + y'^ = 2 ax at 
tan -12. 

14. Find the subtangents and the subnormals of the conic sections and 
the cissoid. 

Ans. Parabola : subt. = 2 x' ; subn. = 2p. 

Ellipse: subt. = {x"^ — a^)/x'; subn. = — h-x'/a?'. 

Hyperbola: subt. = {x"- — a^)/x'\ subn. = W-x'/d^. 

t^- -A 1^ x'(2a — xO - x'2(3a — x') 

Cissoid : subt. = —^ ~- ; subn. = —7 —r- ■ 

Sa — x' {2 a — x'Y 

Find the subtangent and the subnormal of 

15. The hyperbola xy = m. 

16. The semi-cubical parabola ay^ = x^. § 155, fig. 2. 

17. Find the slope of the logarithmic curve x = logay. The slope 
varies as what ? In the curve x = log y the slope equals what ? 

18. Find the equations of the tangent and the normal to x = logaV- 
Show that the subt. = m, and find the subn. 

19. Find the normal, subnormal, tangent, and subtangent of the 
catenary y = ^ (e^/« + e-^/«). §197, fig. 

Ans. -; - e2^/«-e-2^/« ; —^L=; -^ • 

a 4 V2/2 — a2 ^y2 _ a2 

20. The path of a point is an arc of the parabola y^ = 4px, and its 
velocity is v ; fhid its velocity in the direction of each axis. 



VELOCITIES. 29 

Let s denote the length of the path measured from any point upon 
it; then 

ds/dt = V. 

^ _ . dy 2p dx 

From ?/2 = 4 r)X, ^ = t:' 

^ ^ ' dt y dt 

Substituting these values in 

. (r=(iy+(ir. ^'''^- 

we obtain 

dx yv J dy 2pv 

— — and — ~" 



dt V?/2 + 4p2 dt Vy2 + 4p2 

21. Find the velocities required in example 20, when the path is, 

(i) an arc of the circle x^ + y- = r^, 

x^ y^ 
(ii) an arc of the ellipse "^ + y^ — 1» 

X^ 7/2 

(iii) an arc of the hyperbola ~i "" y^ == !• 

22. A comet's orbit is a parabola, and its velocity is v ; find its rate 
of approach to the sun, which is at the focus of its orbit. 

Let p denote the distance from the focus to any point on y~ = 4px ; 
then 

p = x -Jrp. 

.: dp/dt = dz/dt. (1) 

Hence, the comet approaches or recedes from the sun just as fast 
as it moves parallel to the axis of its orbit. 

,dp^d^^ y ^^ ^ 2^ 

dt dt V?/2 + 4p2 

At the vertex ?/ = ; hence, at the vertex dp/dt is zero. 
When y = 2p, dp/dt ={\/2)V2'o. 
When y = ?>p, dp/dt = {p, / Kl) Vl"^ w 

23. The curves y —f(x) and ?/ = F {x) have the point [a, /(a)] in com- 
mon ; show that these curves intersect at this point at an angle whoso 

tangent IS ^^..^y;^- 



30 



DIFFERENTIAL CALCULUS. 



Trigonometric Functions. 

37. A radian is an angle which, when placed at the centre 
of a circle, intercepts an arc a radius in length. It equals 
180° /tt, or 57°.3 nearly. Let u denote the number of radians 
in any angle at the centre of a circle, r the number of units in 
the radius, and 5 the number in the intercepted arc ; then 

u = s/r 5 or if r = 1, u = s. 




38. d*sinu = cosudu. [17] 

d cos u = — sin u du. [18] 

Let the point P (x, y) move along the 
arc XFY of a unit circle. Denote the 
number of linear units in the arc XF 
by s, and the number of radians in the 



F ^ ^ angle XOP by u. We shall then have 



u 

du 



y = sm tc, 
dy = fZ sin u, 



X = cos u. 
dx = d cos u. 



ds, dy = d sin u, dx = d cos ti. (1) 

Angle EDP equals u, and dx is negative ; hence, from the 
triangle UDF by § 33 we obtain 

dy = cos u ds, dx = — sin tc ds. (2) 

Substituting for dy, ds, and dx in (2) their values as given 
in (1), we obtain [17] and [18]. 



39. d tan u = sec^u du. 

Por tan u = 

.*. d tan ic = 



[19] 



sin w/cos u. 

cos u d sin k, — sin tc d cos u 



(gos^u + sin^zO du 



= sechc dii. 



* "When not needed to avoid ambiguity the parentheses after the sign 
d are often omitted. 



TRIGONOMETRIC FUNCTIONS. 31 

40. d cot u = — csc% du. [20] 

For cot u — tan (7r/2 — u). 

.'. d cot u = sec^(7r/2 — u) d (7r/2 — u) 
. = — GSG^u du. 

41. d sec u = sec u tan u du. [21] 

For sec u = 1 / cos ti. 

, sin udu 
.'. d sec u = ^ — = sec u tan u du. 

COS^M 

42. d CSC u = — CSC u cot u du. [22] 
For CSC w = sec (7r/2 — -w). 

.-. d CSC u = sec (7r/2 — u) tan (7r/2 — u) d (7r/2 — ?/) 
= — CSC u cot u du. 

43. d vers u = d (1 — cosu) = sin u du. [23] 

44. d covers u = d (1 — sin u) = — cos u du. [24] 



-f 1. y 



EXAMPLES. 



= sin aa;. dij = a cos ax • dx. 



2. y = cos (x/a). dy = — a-i sin (x/a) dx. 

3. ?/ = cos x^. dy = — 3 X- sin x^dc. 

4. /(^) = t;iu"'6'. /'(^) =: m tan"'-i ^ soc-^. 

5. f{d) = tan 3 0+ sec 3 ^. /'((?) = 3 sec^ 30+ 3 sec 3 tan ; 

6. /(x) = sin (log ax). f{x) = x~^ cos (log ax). 

/(x) changes at the rate of /'(x) to 1 of x (§ 12). 

7. /(x) = log (sin ox). /'(x) = a cot ax. 

8. /(0) = log (tan aO). f\0) = , . "^^ ■ = .^f • 

9. /{()) - log (col aO). f'{0) = — '2 a /sin 2 aO. 



32 DIFFERENTIAL CALCULUS. 

10. fix) = = cos ax — sin ax. 

^ ' sec ax 

11. f{x) = X" e^in x^ f'(x) = x«-i esi° ^ (n + x cos x). 

12. /(x) = sin 7ix • sin"x. f^i'^) — ^ sin»-i x • sin («,x + x). 

13. /(x) = e^ log sin x. /'(x) = e^ (cot x + log sin x). 

14. f{x)= tan (logx). 15. /(x) = log secx. 

16. / (x) = cos X / 2 sin^ x. 

17. /(x) = 4 sin»" ax. f'{x) = 4:am sin*"-! ax cos ax. 

18. /(x) = x^in ^. /'(a^) = ^''° ^ (sin x/x + log x cos x). 

19. /(^) = (sin ^)tan e, f'{e) = (sin sy^^ 6 (i + sec2 e log sin ^). 

20. /(x)=tanai/^. /'(x) = — ai/^sec2ai/^loga/x2. 

21. /(^) = i tan3 d — tan ^ + ^. f'{d) = tan^ ^. 

22. /(x) = e(« + ^)2 sin x. /'(x) = e(« + ^)2 [2 (a + x) sin x + cos x]. 

23. /(x) = e-«2^2 cos rx. f\x) = — e-«-^^(2 a^x cos rx + r sin rx). 

, — (sec Vl — x)2 

24. /(x) = tan Vl - x. f\x) = — ^--— - • 

/'(^) = CSC ^. 



25. /(^) = log^/lzi^2^. 
\ 1 4- cos ^ 



By differentiation derive each of the following pairs of identities from 
the other : 

26. sin 2 ^ = 2 sin 6 cos 6, cos 2 ^ = cos^ 6 — sin2 q^ 

orr • ^ r. 2 tan ^ ^ ^ 1 — tan^^ 

27. sm 2 ^ = — , r- » cos 2 ^ = — , — ■ • 

l + tan26' H-tan2^ 

28. sin 3 = 3 sin ^ — 4 sin^ e, 
cos 3 ^ = 4 cos3 ^ — 3 cos d. 

29. sin {m + n)d = sin mO cos n^ + cos md sin n^, 
cos (m + n) ^ = cos md cos 7i^ — sin md sin n^. 

30. df{a + x2) = /'(a + x2) 2 xdx. 
3L (Z/(ax2) =/'(ax2) 2axc?x. 

dx \a/ \a J a 

33. df {xy) = f\xy) {xdy + ydx). 



INVERSE-TRIGONOMETRIC FUNCTIONS. 33 

Inverse-Trigonometric Functions. 

45. dsin-iuEEdu/Vi - u'* [25] 

That is, the differential of an angle in terms of its sine is the 
differential of the sine divided hy the square root of one minus 
the square of the sine. 

Let y = sm~^u ; then sin y = u. 

du du 

.\dy 



cos y VI — sin^^Z 
du 



Vl — u 



46. dcos-iu = c?/'|-sin-i|A = -— J^^- [26] 



47. dtan-^UEE— ^- [27] 

Let y = tan~^w ; then tan y = u. 

du du du 



.'.dy = 



sec^y 1 +tan^y 1 + ?r 



48. d cot-i u = d(Yv- tan-^ uA =e ^- [28] 



49. dsec-^u- — ^=' [-«^] 

uVu^ — I 

Let y = sec^u. ; then sec // = u. 

du du 



■• <ty 



sec // tan // u \ ir — 1 



* To avoid Uu> ambiguity of thr (loubU> sign t . \V(> shall in thoso for- 
mulas linul, sin ' ((, C(.)s ' u^ otr., [o vahu-s botwrrn aiul t "J. 



34 DIFFERENTIAL CALCULUS. 



50. dcsc-^u = df--seG-Ui\ = ^ • [301 

V2 J uVu^=^ ^ ^ 

51. dvers-iu = — =^=- r311 

V2 u - u^ ■- -^ 

Let y = YeTS~'^u ; then vers y = u. 
du du 



.'.dij 



sin y Vl — cos^?/ 
du 

Vl — (1 - vers ?/)2 



Vl - (1 - i/,)2 V2 u - u^ 

52. dcovers-lu = c//^--vers-IM^ = , ^ [321 

V2 y V2U-U2 ^ -^ 



EXAMPLES. 



, X d(x/a) dx 



a Vl-(x/a)2 Va2 



. dcos-i-= ; dtan— 1-=: 



a 



Va2 _ a;2 ' a a'^ + x2 



, -, X adx , , a; adx 

dcot— 1-= : dsec-i 



a a'^ + x^' a xVx'^ — a^' 

, , X adx , , cc ^x 

d csc-i - = ; — : d vers— 1 - = 



a X V x^ — a 



Vx2 — a2 a V2 ax — x^ 



7nx 



3. y = X sm— 1 ?wx. -^ = sm— 1 mx + 

dX VI — ??22x2 

4. ?/ = tanx tan— ix. -r- = sec2xtan— ^x + :; — , — r- 

dx 1 + x2 



5. y = tan- 



2x d^_ 2 (1 — x2) 

1 + x2 ' cZx "~ 1 + 6 x2 + x^ 



INVERSE-TRIGONOMETRIC FUNCTIONS. 



35 



6. y 

7. y 

8. y 

9. y 

10. y 

11. 2/ 

12. 2/ 

13. 2/ 

14. y 

15. 2/ 

16. 2/ 

17. y 

18. // 

19. ?/ 

20. y 

21. 7/ 



SIR- 1 -^ 

V2 



sec— 1 



1 



2 x2 - 1 
■ cos-1 ,,^ , ^ • 

: tan— 1 (n tanx). 



, x + a 

: tan— 1 

1 — ax 



= sin— 1 vsinx, 

gX g— x 

= cos— 1 — , 

e^ + e-^ 

, Vx + Va 

= tan- 1 -j=- • 

1 — Vax 

= sec- 1 "X • 

\l + x 

^ ^ 1 + Vl + X2 

= cot-1 

X 

= vers-i—j — :• 
1 + x-^ 



■_x 

+ x 



, 8 X — x3 
tan- 1 — ^ 



1 - -} x2 ' 



esc- 1 

2 .i'2 - 1 

1 + x- 



dy ^ 1 

cZx Vl — 2 x~— x2 

dx Vl — x2 

dy _ _ 2wx'*-i^ 
dx ~ x2«+ l' 

cZx cos2 X + n'^sin^ x 

(Zv . _i /sin-ix , loffx \ 

dx V X Vl -x2/ 

cZy^_J 

dx 1 + x2 

dg _ V l 4- CSC X 
dx ~ 2 

dy ^ -2 
dx e-'- + 



Q—X 

1 



# 

dx 2 Vx (1 + x) 

c^y ^ - 1 

dx 2V1 — x2 

dy _ 1 

dx ~ 2 (1 + x2) ' 

# 1 
dx 



•2V-2 






X — X'- 



dy _ 2 
dx 1 + x2 

dy ^ _ 3__ ^ 

d.r ~ 1 +x2* 

'i// -- 2 
(U~ Vl - x-J 

ily_ - 2 ^ 



36 



DIFFERENTIAL CALCULUS. 



t3 + 5cosx 


dy _ 4 


22. ?/ — cos 1 , ^ 

5 + 3 cos X 

9''. ?/ — tan-l ^ 


dx 5 + 3COSX 
dy _ 1 


Vl-x2 


dx Vl - x2 



24. A wheel whose radius is r rolls along a horizontal line with a 
velocity X) ; find the velocity of any point P in its rim, also the velocity 
of P horizontally and vertically. 




The path of P is a cycloid whose equations are 
x = r(^ — sin^), 1 

y — r {\ — cos 6) = r vers ^, / 
where 6 denotes the variable angle DCP, and r the radius CD. 

Since the centre of the wheel is vertically over D, 

V = the time-rate of OD 
= d{rd)/dt = r-dd/dt. 
.-. dd/dt = v/r. 
Differentiating equations (1), by (2) we obtain 

dx/dt= v vers 6 = the velocity horizontally, 
and dy/dt = V'sm6 = the velocity vertically. 



(1) 



ds 
dt 


=V(f/ 


nm 










= V V2 (1 - 


cos e) = v 


^2y, 


/r 






= the velocity of P along itj 


3 path. 




to, 


^ = 0, and 


dx _ d?/ 
dt ~ dt 


_ds. 
~ dt' 


= 0. 




.E, 


"=!' 


dx _ dy 

dt ~ dt ' 


= ^, 


ds_ 
dt 


vV2. 


.K, 


e=7r. 


dx _ ds _ 
dt ~ dt~ 


= 2.1), 


dy _ 
dt 


0. 



(2) 

(3) 
(4) 



(5) 



THE CYCLOID. 37 

From (5) we obtain 



ds/dt : V = V2r -y : r. 

Hence, the velocity of P is to tliat of C as the chord DP is to the 
radius DC ; that is, P and C are momentarily moving about D with 
equal angular velocities. 

26. Find the subnormal and the normal of the cycloid. 

Subn. = r sin d = PH = ED. 

Thus the normal at P passes through the foot of the perpendicular 
to OM from C. Hence, to draw a tangent and normal at P, locate 
C, draw the perpendicular DCB equal to 2 r, and join P with B and 
D ; then PB and PD will be respectively the tangent and the normal 
at P. 



Normal = DP = ^DB • DH = V2 



r-y. 



26. Eliminating 6 in equations (1) of example 24, find the equation of 
the cycloid in the form 



x = r vers-i (y/r) =ji V2 ry — y^, 

27. The equation of the tangent to the cycloid is 

y — y'= ^i2r — y')/y' (x — x'). 

28. A vertical wheel whose circumference is 20 ft. makes 5 revolutions 
a second about a fixed axis. How fast is a point in its circumference 
moving horizontally when it is 30° from either extremity of the horizontiil 
diameter ? ^,^5^ 50 ft. a second. 

29. AVhat is the slope of the curve y = sinx ? Its inclination lies be- 
tween what values ? What is its inclination at x = 0? What at x = 
7r/2? 

The slope = cosx; hence, at any point, it nuist bo something be- 
tween — 1 and + 1 inclusive. Hence, the inclination of the curve 
at any point is sometlung between and 7r/4, or something between 
3 7r/4 and tt inclusive. 

30. Find the ecjuation oi the tangent io the curve // = sin.r ; ij =■ tana- ; 
y = sec X. 



38 DIFFERENTIAL CALCULUS. 

MISCELLANEOUS EXAMPLES. 

1. y = log tan— i x. 10. y = e«^ sin"* rx. 

2. y= {x + Vl — x2)« 11. ?/ = log {log {a + 6x«)}. 



3. y=J'-^' 

\ (1 + X2)^ 



^ Vl + X^ + Vl - X2 

(i + ^¥ ■ ^ vrr^^- vr^^" 

4. ?/ = ^^ — In example 12 rationalize the 

^ ' denominator before differentiating. 

5. y = e^ta,n-^x. 

S + 2X 13. /(x) = (a2 + x2)tan-i^- 

6. ?/ = sin-i — ;=-• " 

Vl3 

7. /(x) = e^" + ^)2sinx. 



14. ?/ = Vl — x^ sin— 1 X — X. 



- X log X . . .^ . 

8. 2/ = y^f^ + log (1 - X). 



15. y=rtan-i-+ log^/^~^. 
^ \x + a 



/I? 



_ , a ,^ vx2+ 1 +x 

9. y = sec-i . • 16. y = , 

Va2 — x2 Vx2 + I — X 



Vx2 + a2 + Vx2 + 62 
17. y = . = , 

Vx2 + a2 — Vx2 + 62 



^^_2^_/2+ / x2 + a2 /x2_+_6_2\ 

c?x a2-62V \x2+62 \x2 + a2/ 

18. 2/ = log(Vl + x2+ Vl-x2). ^ = -(l--=L^)- 

dx x^ Vl — x*/ 



19. f{x) = {x-S) e2^ + 4xe^ + X + 3. 

20. y = \og {2X — 1 + 2 Vx^ — x—1). 



dx Vx2 — X — 1 



21. 



^ = log(^) 



l + x\i/4 tan-ix (Z?/ x2 



dx 1 — x^ 



■T— ?/ 

22. x = e y . 



dx X (log X + 1) 



X — ?/ 
Here log x = • 



23. y 



MISCELLANEOUS EXAMPLES. 39 

X2 



1 + -^ 



1+ "" 



1 + etc. to infinity. 

XT ^^ 



1 / f /^^ ^x . 1 ^ cZ?/ 1 /x + a 

24. ?/ = log(x 4- Vx^— a^) + sec-i -• -t~ — ~\\ 

°^ ' a dx x\x — a 



, Vl - x2 + X V2 (^2/ V2 

25. ?/ == log , — = , , -p 

V 1 - x-^ dx ( VI - x2 + X V2) (1 - xP) 

oa 1 ^/vTTx2 + x dy 

26. y = \og\ -^ 

^ Vl + x2 - X dx 



dx VI + x2 
Here ?/ = log( Vl + x^ + x). 



o^ 1 /2x2-2x+l , ^ - 



2x d?/ _ 8 x2 

-2x2* (Zx~ 4x^ + 1 



,,0 ^1^11 /x — a dy 2ax^ 

28. ?/ = cot- 1 - + log "X — I f^ = -T 1' 

X ° \ X + a cZx X* — a-* 

__ . , X tan ^ d?/ a- tan 6 1 

29. ?/ = sin-i -^ = 



Va^ — X- dx a- — X- Va- — x- sec- ^ 



i /x-2\3/i) . dv X- — 1 



CHAPTER III. 
PROBLEM OF EATES SOLVED BY LIMITS. 

53. Limit. When according to its law of change a vari- 
able approaches indefinitely near and continually nearer a 
constant, but can never reach it, the constant is called the 
limit of the variable. 

It is assumed that the reader is familiar with the elementary theorems 
of Limits ; but for convenience of reference we state them below : 

1. If two variables are equal, their limits are equal. 

2. The limit of the sum, or product, of a constant and a variable is 
the sum, or product, of the constant and the limit of the variable. 

3. The limit of the variable sum, or product, of two or more variables 
is the sum, or product, of their limits. 

4. The limit of the variable quotient of two variables is the quotient 
of their limits, except when the limit of the divisor is zero. 

54. Notation. The sign = denotes "approaches as a 
limit." Thus, Aa; = is read " Ax approaches zero as its 
limit.'' 

The limit of a variable, as z, is often written It (z). 

* ^^.\ T^ L or It -^? denotes It ( -^ ) when Ax = 0. 
Ax := |_Ax J Ax \Axy 

Let v', x', and y' denote any corresponding values of v, x, 
and y, from which \v, Ax, and A?/ are estimated. Let the 
rate of v be the unit of rates ; then evidently 

Ay _ r the v-rate of y at some value of 3/ 1 

^v '^ between y' arid y' -j- A?/ j * ^ ^ 



LIMIT OF AT/ AX. 41 

.'.It (A?//Av) = the v-rate of y at the value ij. (2) 

Also It (Ax/Av) = the v-rate of x at the value x'. (3) 

Dividing (2) by (3), we obtain, in general. 



limit Fa?/"! _ the v-rate of y 



mit rA|/"| 

= oLaxJ 



Ace = Aa? the i7-rate of 



X 



(4) 



Comparing (4) with (1) of § 12, we obtain [33]. 



To illustrate (1) and (2), let s denote the number of feet a falling body- 
descends in t seconds. Let s' and f denote any corresponding values of 
s and t, from which As and A^ are reckoned ; then, evidently, 

As _ / the time-rate of s at some value of s 1 
A^ I between s' and s' + As J 

.'. It {As/ At) = the time-rate of s at the value s\ 



EXAMPLES. 

By § 53 and [33] of § 55, prove 

1. d {uy) = ydu + udy^ or formula [7]. 

Let z — uy, and let x' represent any value of x, and u\ y\ and z' 
the corresponding values of u, ?/, and z, respectively ; then 

z' = uy. (1) 

When X = x' + Ax, u = u' + A?,(, y = y' + At/, and z = z' -\- Az \ 
hence, 

z' + Az= {uf + An) (//' + Ay) 

= li'y' + y'Au + u'Ay + AuAy. (2) 

Subtracting (1) from (2), we obtain 

Az = y'Au + u'Ay + AuAy. (3) 

= ?/'lt^ + lt(«' + AiO It ;-^- 

Ax ^ A.r 

dx dx dx J L J 



42 DIFFERENTIAL CALCULUS. 

Hence, as x' is any value of x, we have, in general, 
dz = d [mj) = ydu + udy, or [7]. 

If in [7] we put for ?/ the constant a, we obtain 
d{au) = adu, or [4]. 

If in [7] we put viv for u, we obtain 

d {vwy) = loydv + vydio + vwdy^ or [8]. 

2. d (XL + ?/ + 2 + «) = dii + dy + (^2, or [3]. 

Let v = 11 + y -\- z-\- a; 

tlien Av = Ah + A?/ + A^, 

, Au , Am , , Ay , , Az 
.-. It — = It " + It -^ + It — • 
Ax Ax Ax Ax 

dv _ du dy dz ^ 
' ' dx dx dx dx 

.-. dv = d {u + y + z + a) = du + dy + dz. 

3. lly = z, dy = dz, or [1]. 

If y = z, Ay = Az. 

Ax Ax' ' ' dx dx 

4. Aa = 0; .-. da = 0, or [2]. 

5. d{u/ij)-{ydii — udy)/if-, or [9]. 

Let v= u /y \ 

u' + A?( u' y' Au — u' Ay 

then Av = , , ^ ; = ~:rx — VT^ ' 

y' + Ay y' y'^ + y' Ay 

6. Assuming the binomial theorem, prove : 

d{iO') — nvo^-^du^ or [13]. 

56. By [33] of § 55 the problem of rates is reduced to one 
of limits. By theorems proved later in this chapter, proofs 
by limits are very much abbreviated. 



INFINITESIMALS. 43 

The reader should note that we cannot write 

Ay^lt(Ay)^0 

Ax-lt(A2^)-0 ^^ 

For any proof of the principle applied in (1) fails vjhen the 
limit of the diviso?' is zei'o. Moreover, the determinate expres- 
sion lt(A?//Aa?) cannot be identical with the indeterminate 
expression 0/0. 

To find It (A?//Aa^), we find the limit of an equal variable, 
as in the examples of § 55 ; or we find the limit of some 
variable which, though not equal to A^z/A^c, has the same 
limit, as in § 63. 

The theory of limits never gives rise to the form a/0, or to 
any of the indeterminate forms 0/0, etc. 

57. Derivative as a limit. By § 29 and [33] we have 
It (A?/ /Ax) = the derivative of y with respect to x. 

Or, since A/(£c) =f{x-\- Ax) — f(x), we have 

It (A?// Ax), or dy / dx, is often denoted by D^ij. 

58. Infinitesimals. Zero is defined by the identity 

a — a = 0. 

An infinitesimal is a variable whose limit is zero. 
Hence, Ax = may be read "Ax is an infinitesimal." 
In approaching its limit zero, an infinitesimal becomes in- 
definitely small and continually smaller, but it never equals 
zero. Any small quantity becomes an infinitesimal when it 
begins to approach zero as its limit. mA when it reaches any 
particular degree of s})iallncss. A (luantitv, liowevtn- small. 
which does not approach zero as its limit is ncU an intinitesi- 
mal. 



44 DIFFERENTIAL CALCULUS. 

Infinitesimals are indefinite variables used as auxiliaries in the study of 
finite quantities. Their essence and utility lie in their having zero as their 
limit, and not in their smallness. The reader should not make the study 
of them difficult and obscure by thinking of them as mysteriously small. 

59. An infinite is a variable which under its law of change 
can exceed all assignable values, however great. The general 
symbol for an infinite is oo . The reciprocal of an infinitesimal 
is an infinite, and conversely. 

For example, when x = 0, l/x = oo; that is, when x is an infinitesi- 
mal, 1 / X is an infinite. 

When ^ ^ 0, cot {±6) = ± co , and tan (7r/2 qp ^) = ± co . 
When X = 0, log x= — cc . 

An infinite does not approach a limit ; in arithmetic value 
it increases without limit. 

60. Geometric meaning of It (Ay /Ax). Let mn be the 
locus of y =f{x), FF' a secant, and FD a tangent at P. Draw 

the ordinates MF and NF', also 
P(7 parallel to OX 

Let 0M= X, and MJST = Ax 5 
then MF = y, and CF^ = Ay. 

Hence, Ay / Ax= Cr / FC = i\\Q 
slope of the secant FF'. 

Conceive the secant PP' to re- 

X 

volve about P so that 

arc FF' = ; then Ax = 0, Ay = 0, ■ 

and the slope of the secant = the slope of the tangent at P. 

Hence, It {Ay / Ax) = the slope of the curve y = f(x) at 
the point (x, y). 

CoR. If when Ax = 0, Ay / Ax varies, the locus of y =f(x) 
is a curved line, and in general Ay /Ax approaches a limit ; 
but at a point where the locus is perpendicular to the cc-axis 
Ay I Ax = 00 when Ax = 0. 







PRINCIPLES OF LIMITS. 45 

61. The limit of the ratio of an infinitesimal arc of any 
plane curve to its chord is unity. 

Let s represent the length of the arc mP (§ 60, fig.); and 
arc PaP^ = As ; then PC = Ax. 

Since 5 is a function of x, we have 

It (As / Ax) = ds/ dx. (1) 

But It (chord PP'/Ax) = It (sec CPP") 

= sec BPD = ds/dx. (2) 

Dividing (1) by (2), we obtain 

li^it r As n 

As^oLchordPP'J * ^^ 

Cor. Let it and s have the same meanings as in § 38 ; then 

limit r ?^ "I _ limit f 2s ~| _ limit p • XP l _ 
u = Lsin i^ J ~ s = 1_2 sin uj ~ XP = o\_2- CP_\ ~ ' 

That iSj the limit of the ratio of an infinitesimal angle to its 
sine is unity. 

62. The limit of the ratio of tiuo variables is not cJianrjed 
tuhen either is replaced by any other variable the limit of whose 
ratio to it is unity. 

Let a, ai, /?, and /3i be any four variables, so related that 

lt-=:l, ltf = l, and \tj = c,. (1) 

a a ai ^1 _ tti a /8i 

IS A «1 (S /3\ * ^ 

in wliich a is r('})l;u'od by a,, and (S by /)',, witlKnit i-haui^ing 
the limit. 

This principle^ o\'[c\] tMinMcs us to sini|ilifv a pnU^lciu o\' limits l\v sub- 
slitutinu- for an ii)tinitisiinal :\vc its I'lionl. as in ilic t'oUowiui;' articK>. 



46 



DIFFERENTIAL CALCULUS. 



63. Subtangent, subnormal, tangent, normal, in polar 
curves. Let arc mP = s, and arc PD = A.5 ; then APOD = A^, 
circular arc PM= p\9, and MD = Ap. Draw the chords P3I 
and PD, the tangents RPH and TPZ, and ZH perpendicular 
to PH, Z being any point on the tangent PZ. 

When As = 0, the limiting positions of the secants PM and 
PD are the tangents RPH and TPZ, respectively ; hence, 

It (Z PMD) = Z PiPK = 7r/2 - Z PHZ, 

It (Z ODP) = Z OPT = ,/. = Z ^^-P, 

and It (Z MPD) = Z ^P^. 




Again, by §§ 61 and 62, and trigonometry, we have 

si n JiPD 
sinPJ/D' 



It^^^lt ^^ 
A.5 chord PD 



It 



ds 



sin 7ZPZ 



PZ' 



(1) 



POLAR CURVES. 47 

, pAO .choid MP , sin MDP 
Also, It '^—— = It — — ^ -^^ = it -. — ^. .,,■., ; 

' As chord FP sm PMP 

,..^^.3ini.ZP = f|. (2) 

From (1) and (2), it follows that, if 

ds = PZ, dp = HZ and pdQ = HP. 

Draw OT perpendicular to OP, and PA and ON perpen- 
dicular to the tangent TP. Then the length PT is called the 
polar tanrjent • PA, the polar noi^vial ; OA, the polar subnor- 
mal; and OT, \he polar suhtangent. 

From the right-angled triangle HPZ we have 



ds" = dp' + pV/^'-^ ; 


(3) 


. , pdd , fh . , pdO 
8\\\\l/ = ^——i cosiA = -r' tan i/^ = ^-- — 
^ ds ^ ds ^ dp 


(4) 


Polar subt. = OT = OP tan i// = phlO/dp. 


(5) 


Polar subn. = OA = OP cot </^ = f/p/(/^. 


(6) 


/ iff^ 




Polar tan. = PT = ^OP'+OT' = p V^ + p' " ' 


(7) 


1 J ■> 




Polar norm. - ^P - VOP- -f 0^1=-' - Vp- + ' ,^.. 


(8) 


j9 = 0N= OP sin ,// = p-dO/ds 




P' 


/0^ 


^p'-{-{dp/doy 


(.n 


cj> = xlj + 0. 


(10^ 



Cor. Tf PZ represents tlu^ vi^h^city at /' of (^p. ^A ;doiig the 
line of its path, HZ and /*// will rt>prosont its i-omponont 
velocities at /* along the radius vtH'tor and a line perpen- 
dicular to it. 



48 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Find the subtangent, subnormal, tangent, normal, and p of the 
spiral of Archimedes p = ad. 

Ans. subn. = a ; subt. = p~/a ; norm. = Vp2"+~a^ ; 

tan. = p Vl + pVa^ ; P = p'"/ {p^ + a^)!/^. 

2. In the spiral of Archimedes show that tan xj/ = 6. 

3. Find the subtangent, subnormal, tangent, and normal of the 
logarithmic spiral p = a^. 

Ans. subt. = p/loga; subn, ^ploga; 

tan. = p vT+ (loga)-2 j norm. = p Vl + (loga)'-^. 

4. In the logarithmic spiral, show that xj/ is constant. 

If a = e, t/' = 7r/4, subt. = subn., and tan. = norm. 
Since the logarithmic spiral cuts every radius vector at the same 
angle, it is often called the equiangular spiral. 

5. Find the subtangent, subnormal, and p of the lemniscate of 
BernouilU p^ = a^ cos 2 6. 

Ans. subt. = — p3/of2 sin 2 (9 ; subn. = — a^sin 2 6 /p. 
p = pV Vpi-^a^sm^2e = p^/a^ 

6. In the lemniscate show that -^ = 2 ^ + 7r/2 and = 3 ^ + 7r/2. 

7. In the curve p = a sin^ (d/S), show that (p = 4x1/. § 155, fig. 14. 

8. In the parabola p = a sec^ ((9/2), show .that (p + xp = 7t. 

9. The velocity of a point along the spiral p = ad ifi v ; find its com- 
ponent velocities along the radius vector and a line perpendicular to it. 

10. By the method of limits {§ 60, fig.) prove that, if PB = dx, BB = dy 
and PB = ds, where s = mP. 

64. Orders of infinitesimals. Any variable whicli is 
neither an infinitesimal nor an infinite is called a finite 
variable. 

Two infinitesimals are said to be of the same order when 
their ratio is a constant or a, finite variable. 



ORDERS OF INFINITESIMALS. 49 

For example, wlien Ax == 0, 7 Ax and (5 a + G)Ax are infinitesimals of 
the same order ; so also are 9 (Ax)^ and (8 x — 7) (Ax)^ ; so also are Ay 
and Ax when It (A?/ /Ax) is other than zero. Again, when 6 =71:/ 2, 
1 — sin ^ and cos^^ are infinitesimals of the same order ; for 

limit r l — sin d ~\ _ limit r 1 ~l _ 1_ 

e = 7t/2 L cos'^^ ]~d=7t/2\_l + sin ^J ~ 2' 

In order to classify the infinitesinials in any problem as 
belonging to different orders, we choose some one of them as 
the principal infinitesimal, and adopt the following definitions: 

An infinitesimal is of the first order when it is of the same 
order as the principal infinitesimal ; of the second order when 
it is of the same order as the square of the principal infinitesi- 
mal ; and so on. 

In general, an infinitesimal is of the nth. order when it is of 
the same order as the nth power of the principal infinitesimal. 

Thus, when Ax is taken as the principal infinitesimal and It (Ay /Ax) 
is other than zero, Ay is an infinitesimal of the jirst order ; 5 x (Ax)- and 
1 yAijAx are infinitesimals of the second order; 7?/(Ax)3 and 4x(Ax)-A?/ 
are of the third order; and ?/(Ax)'* and x'-(Ax)'*-"* (A?/)'" are of the >itli order. 

65. Notation. Let v^, V2, ' • •, v^ represent finite variables 
or any constants except zero, and let i represent the principal 
infinitesimal ; then v-^i, v^i^, • • •, vj" will represent respec- 
tively infinitesimals of the first, the second, • • •, the ??th order. 

According to this notation, Ax = i is road " Ax is the principal infini- 
tesimal " ; 5xAx = i;i is read "HxAx is an infinitesimal of the tirst 
order"; 7xAxA// = Ua''" i« read "7xAxA// is an intinitcsimal of the 
second order"; and so on. The subscript uchhI not be wriHtu w'nh 
the first V which a|)pcars in any i)robUMn or discussion. 

A. finite quantity may be regardi^l as an inlinitosinial of tl>e 
zero order ; for x = xi^ and 'vl^ n r. 

Ex. \iO= L 1 — cos = vi-. 



limit r l — cos $ -\ _ limit r l — c os <?~i 
<^ = L ^"-^ J ~ f^ = L sin-(~J 

— ^'"^'' r ^ - "1 — - 






50 DIFFERENTIAL CALCULUS. 

J) 66. Geometric illustration of in- 
finitesimals of different orders. 

Let CAB be a right angle inscribed in 

the semicircle CAB^ BB a tangent at 5, 

and AE a perpendicular to BB. From the 

•" similar triangles CAB, BAB, and AEB we 

have 

AB:AB = AB:AC; /. AB = (1 /AC) aW ; (1) 

and BE :AB= AB :BC; .: BE = (l/BC) -AB - AB. (2) 

Suppose A to approach I?, and let AB = i ; then, from (1) and (2), 




AB={l/AC)-P 



Vl" 



and BE = {l/BC) - vi^ • i - v^i\ 

67. Orders of products and quotients. 

The order of the 'product of two or more infiiiitesimals is equal 
to the sum of the orders of the factors. 

For vi'''vj"' = vvj'' + '^. 

The order of the quotient of any tivo infinitesimals is equal 
to the order of the dividend minus the order of the divisor. 

For vi^/vj''' = (y/v^i''-'^. 

68. If the limit of the ratio of one infinitesimal to another is 
zero, the first is of a higher order than the second; and con- 
versely. 

For It (vi^'/vj"') = It [(v / ?^,„) i'*-''*] = 

wlien^ and only when, n > m. 

Cor. If the ratio of one infinitesimal to another is infinite, 
the first is of a lower order than the second. 

69. If the limit of the ratio of tivo variables is unity, their 
difference is an infinitesimal of a higher order than either ; 
and conversely. 



LAWS OF INFINITESIMALS. 51 

Let a = (3 -^ €', B '^ ^'^ "S" 

then lt(a/;8) = l+lt(€/^). . ^-.->^-... > 

If lt(a/^) = l, lt(e/^) = 0; 

hence, by § 68, e is of a higher order than y8. 

Conversely, if e is of a higher order than /3, 

lt(e/^)=0; .\\t{a/P)=l. 

CoR. If arc PP' = i, arc PP' = chord PP' + vi% where 
Also, if angle u = i, u = sin u + vi'% where 7i'> 1. § 61 

70. From sums of infinitesimals of different oi'dei's, all infini- 
tesimals of the higher orders vanish in the limit of a ratio. 

., vi + Voj^ + v^i^ + • • • ..vi 
For It-y— — |-r^- — ^^^- = lt — • §62 

This principle of limits often greatly shortens the opera- 
tion of finding the limit of a ratio, and together with [oo] of 
§ 55 furnishes the following simple 

71. Rule for differentiating a function. 

(1) Pind the vahce of the increment of the f miction in terms 
of the increments of its variables. 

(2) Sup2)osi7ig the increments to he infiniteslmah of the first 
order, in all sums drop the infinitesimals of the hif/Iicr oi'drrs. 
and in the remaining terms substitute differentials for incrc- 
ments. 

For by § 70 the infinitesimals of the higher orders in a sum 
will vanish in tlie limit, and by [oo] of -J 55 differentials will 
take the place of incriMuents in the remaining (eims. 

Cou. Anticipating, in sti>p (1\ the resnlt o{ ■>[('y K^-^ ^'^'^' 
need to express exactly only tliosi^ tciaus o{ Ay^//"! \\ Inch arc 
linear in \u. (Sei> (^xaiuplrs 7 {\^ 



52 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

u, y, ' • ' being different functions of x, by § 71 prove 

1. d {uy) = ydu + udij, or [7]. 

A {uy) = {u + All) (y + A?/) — uy § 8, example 3 

= yAu + uAy + AuAy. 

Let Ax = i; tlien AuAy = vi'^. Hence, dropping AuAy, and sub- 
stituting d{ui/), du, and dy, respectively, for A{uy), Am, and Ay in 
the remaining terms we obtain [7]. 

2. d (u/y) = {ydu - udy)/y\ or [9]. 

/M\ _ M + Au _u _ yAu — uAy _ 
\?//y + A?/ ?/~ ?/2 + yAy 

if. — -yio^ and yAy — Vii ; hence, in the sum y- + yAy, by (2) of w^ / 
§ 71, we drop yAy. Substituting differentials for increments in the {a c.CM( 
remaining terms, we obtain [9]. V i- }j^i V^X Ci/u.<^ ijUf^t! 

en. cKjtr^i.- (J^\) 
S. d{au) = adu, or [4]. 5. Formula [8]. 

4. Formula [3]. 6. d{u'') = nu''-'^du, or [13]. 

A (m") = {u + Aw)« — u'» ^ 

= nu^-^Au H ^jT — ^ M»--Ait2 + • • .. 

.'.d{u^) = nu'^—'^du. 

7. dsmu= cosudu, or [17]. 

A (sin u) = sin {u + An) — sin w 

= cos u sin Au + sin u cos Am — sin u by Trig. 

= cos u {Au — ui") — (1 — cos Au) sin u. § 69, Cor. 

= cos It • Au — vi" ■ cos u — 152^" sin u. § 65, example 

.-. dsinu = cos udu. 

In obtaining the value of A (sin it) we express exactly only those 

terms which are linear in Aw ; for by § 70 all the other terms vanish 

in the limit. 



8. cZ cos u = — sin M(^M, or [18]. 9. ds — VdxP^+df. 

Let As = arc PP' =i; § 60, fig. 

then As = c hord PP' + ui«, where n > 1, § 69, Cor. 

= V Ax^ + A2f + vi«. 
.'. ds = ^dx?- + dy"^. 



DIFFERENTIAL OF AREA. 



53 



10. Find the differential of tlie area between the x-axis, the curve RP^ 

or y = f{jc), the fixed ordinate HR, and tlie variable ordinate MP, or y. 

Let P be any point (x, y) on the 

curve. 

Conceive the area HRPM as gener- 
ated by the ordinate JlfP, or ?/, and 
denote this area by A. 

Let MB = Ax ; then Ay = DP\ 
and AA = MBP'P = yAx + PDP'. 
Let Ax = i ; then, since 

PDP'<AxAy, PDP' = vi^ 
Hence, AA = yAx + vi^ ; 

.-. dA = ydx. 
If dx = MB, dA = MBBP. 

From (1) the x-rate of ^ is ?/ to 1 of x. 




B X 



(1) 



11. Find the differential of the area of a polar curve. 
Let OB be any fixed radius vector, 
and P any point (p, 6) on the curve BPh 
referred to the pole O and the polar axis 
OX. Conceive the area BOP as gener- 
ated by the rotation of the radius vector 
/3, and denote it by vl. 

Let /_ POP' — A9, and draw the cir- 
cular arc PB with as a centre ; then 

Ap = BP\ pAe = PD, 
and A.4 = OPP' = OP I) + DPP' 

= ip-At)+ DPP\ 
Let Ad = i; then, since DP'P < pAO • Ap, 

DPP' = vi-. 
Hence, AA = ^p-AO + vi- ; 

.-. dA = )rp-dtK 
If dd= Z POP', dA = (he i-iivuiar sortor OPD. 
From (1) the ^M-ate of .1 is p~/2 io 1 of 0. 




(1) 



12. rnni^ (he (Iummhmu in § 7() by § T)."). 

13. 11" an inliniU'sinial he mull ipliinl ov 
the order o^ (ho in(ini(i>siui;il is uol ohanL: 



ili\ idi'il by any (inite quantiiy. 

■a. 



54 DIFFERENTIAL CALCULUS. 

72. Orders of infinites. The reciprocal of an inlinitesimal 
of any positive order is an infinite of tlie same order ; hence, 
the different positive orders of infinites may be regarded as 
negative orders of infinitesimals ; and conversely. 

According to the notation in § 65 we may write the different 
positive and negative orders of infinitesimals as below: 

v_ni~'\ ' ' -, v_2i~^, v_l^~^ vi^, v-^i'^, v^P, : • • vj"", (1) 

where the negative orders of infinitesimals are positive orders 
of infinites. 

Let c/D = ^~^ ; then the series in (1) becomes 

where the negative orders of infinites are positive orders of 
infinitesimals. 

Infinitesimals and infinites of all orders are variables whose 
general laws of combination are the same as those of finite 
quantities. 

An infinite or an infinitesimal of the zero order is a finite 
quantity. 

Cor. The product of an infinitesimal and an infinite of the 
same order is a finite quantity ; for 

When i is indefinitely small each term in (1) is indefinitely 
small in comparison with the one which precedes it, but indefi- 
nitely large compared with the one which follows it. 

73. 00^^ -0 = 0, where-n is any 2'>ositiv6 number. 

Let cf^i"' = v, (1) 

where neither v nor its limit is zero, and m is any positive 
number. 

Since ^'"'' -0 = 0, it evidently follows that 
oc« • = 00" (^•"'" 0) = (oc &Y • 

= ?;« • = 0. by (1) 

By Cor. of § 72, oo in (1) denotes an infinite of the ??ith order. 



THE EXPRESSION A/0. 55 

Cor. 00^^=1 and 1-=° = !. 

For log (cc") = log GO 

= 0-00 = 0; 
.-. ^J = l. 
Again log (1 * '^) = ± go log 1 

= zboo-0 = 0; 
.-. 1"- = 1. 

74. op, or absolute infinity. The expression a/0 fre- 
quently occurs in mathematics. The question arises, What 
does the expression a/0 (written as one symbol op, which is 
read ' a-by-zero ') symbolize ? 

Any power of an infinite expresses no part of a/0 as a 
quotient ; for, by § 73, co" into the divisor equals 0, or no 
part of the dividend a. Since ap symbolizes that of which 
no part can be expressed by any power of a mathematical 
infinite, it must symbolize that which transcends all mathe- 
matical quantity, or absolute infinity, of which we can have 
no positive idea. 

Since ap is not a mathematical quantity, it is not subject 
to mathematical laws, and the expressions 

ap / ap, ap • 0, ap — ap, ap^, 1^ 
are indeterminate forms. See Chapter V. 

We would naturally conclude that 

2/0 = 2(1/0), 3/0-3(1/0), • ••. 
But tan(7r/2) = 1/0, or 2/0, or 3/0, • • •. 

Tlie inconsistency of these results illustrates the imiiossibility of reason- 
ing:; with the symbol ap. 

Sometimes when one of two related variables assunu^s the form ap. we 
know the value of the other. 

For example, when tan (f> = ap, i.e. when tan assumes the form ap, 
wc know that is eoterminal Nvith 7r/2, or 3;r/2 ; and conversely. 

Wlien cot = ip, wc know that is ci)tenninal with ov ,t. 

If, when x = (', f(.r) z_0 /a _0, the reciprocal of/^.r^ as- 
sumes the form fz/O, or o/k wIumi .r = c. 



56 DIFFERENTIAL CALCULUS. 

75. Ill this chapter, to obtain the ratio of differentials, or 
the ratio of rates, we employ infinitesimal increments of 
variables as auxiliary quantities. The division of infinitesi- 
mals into orders affords clear and brief statements of prin- 
ciples of limits which greatly abridge and simplify the work 
of finding the ratio of differentials. In this as in the previous 
chapters, differentials are regarded as finite quantities. 

76. Limit in position. When according to its law of 
motion a point, or line, approaches indefinitely near and con- 
tinually nearer a fixed point, or line, but can never reach it, 
the fixed point, or line, is called the, limit of the variable 
point, or line. 

For example, when in § 60 arc PP' = 0, tlie fixed point P is the limit 
of P', and the tangent PD is the limit of the secant PP\ 

Whether the word limit has reference to magnitude or to 
position will always be evident from the context. 

EXAMPLES. 

1. When X = c, a/ (x — c) = ± co ; when x = c, a/ {x — c) = ap. 

2. When x =0, a/x = ± oo ; when x = 0, a/x = ap. 

3. When = 7r/2, sec = ± oo ; when = 7r/2, sec cp = ap. 

4. When 0=0, esc = ± oo ; when = 0, esc = ap. 

a/(p{x) ^ fix) 

a/f{x)-cp{xy ^^ 

and /(x)-[a/0(x)]=a/(x)/0(x). (2) 

If /(c) = (c) = 0, (1) and (2) become respectively 

c^/c^=0/0, and -0^=0/0. 
Hence, any expression in x which assumes the indeterminate form 
ap/ ap or • ap for any value of x can be so transformed as to assume 
the form 0/0 for the same value of x. 

6. When Ax = 0, for what points on the locus of 2/ =/(x) is A?/ /Ax 
infinitesimal ? infinite ? finite ? 



CHAPTER IV. 
SUCCESSIVE DIFFEEENTIATION. 

77. Successive differentials. The differential of du is 
called the second differential of u-, the differential of the 
second differential of u is called the third differential of u ; 
and so on. d {du) is written d'^u ; d (d'^u), or d d du, is written 
d^u ; and so on. The figure above d denotes how many times 
in succession the operation of differentiation has been per- 
formed, dti, dhi, d^u, • ' -, d'^^u are called the successive differ- 
entials of u. 

The differential of an independent variable, being arbitrary, 
is supposed to have the same size at all values of the variable. 

Hence, when (as in this chapter) x is independent, dx is to 
be treated as a constant in successive differentiation. 

Ex. Find the successive differentials of u when u = az^. 
du = 4 ax^dx ; 

d^u = i adx ' d (x/^) = 12 ax-dx"^ ; 
dhi = 12 adx^ • d {.x'-) = 24 axdx^; 
dhi = 24 adx-^ ; d/'ii = 0. 

Note that dhi ~ ddu. ; dit- zz {du)- ; d (u-) ~: 2 udu. 

EXAMPLES. 
Find dii^ d-u, and d'ui, wIumi 

1. u = r).r"' + 2 .r- — '■).(•. d'ti --' '.\0 d,r\ 

2. u = (.r- — (').)• + 12) ('■'■. (/■'// = .r-(^'( /.(••'. 

3. u = .r'\oixx. d^u - 2.V ULfK 

4. u = loi; sin .r. d-^'u = 2 ros.r sin - • .•'••. 



58 DIFFERENTIAL CALCULUS. 

5. u = tan X. dhi = (6 sec% — 4 sec^x) dx^. 

6. ?/ = log ax ; find rf-^?/. d^y = — 6 x- ^ cZx^. 

7. y = e— -^ cosx ; find #?/. d^y = — 4 e-^ cosxdx*. 

8. y = e*' sin x ; find d'^y. d'^y = — 8 e^ cos xcZx^. 

78. Successive derivatives. The derivative of the first 
derivative of a function is called the second derivative of the 
function ; the derivative of the second derivative is called the 
third derivative ; and so on. 

When X is independent, 



d du d'^u d dhc 


dhc 


d d^'-'^u d'ht 


dx dx ~ dx^ dx dx^ 


= dx' ' 


' dx dx^'^^ ~ dx"" 



Dividmg by dx^ both members of the answers to examples 1-5 in § 77, 
we obtain in each case the third derivative of u with respect to x. 

The successive derivatives of /(^) are denoted by 
f'(x), f\x), f"'(x), r\x), ' . •, f\x). 

Thus if /(x) = xS /' (x) = 4 x^, f' (x) = 12 x% 

f'" (x) = 24 X, p^ (x) = 24, /v (X) = 0. 

Hence, if u ^^ fix) and x is independent, 

Successive derivatives are often called successive differential 
coefficients. 

79. The nth derivatives of some functions can be readily 
obtained by inspection. 

Ex.1. /(x) = e^; find/«(x). 

r{x) = e-, nx) = e-, r'{x) =e-, • • • ; .-./"(x) = e-. 
Ex.2. f{x) = a^; find/»(x). 

r (X) = log a • a^ r'(x) = (log a)^a-, r'{x) = (log a)^a^, - • -; 
.: /«(x) = (log a)''a^. 



SUCCESSIVE DERIVATIVES. 59 

Ex. 3. fix) = log (1 + x) ; find/«(x). 

r{x) = (1 + x)-i, r{x) = (- i)(i + x)-2, 

r-{x) = (- 1)2 |2 (1 + x)-3, /iv(:c) = (- 1)=^ ^ (1 + x)-\ ■■■; 
.■.f>-{x)={-iy^-'^ \n-l {l +x)-« 
Ex. 4. /(^) = cos cW ; find/»(^). 

/'(^) = — a sin a^ = a cos {ad + 7r/2), 

/''(^) = — a-^sin {a9 + 7t/2) = a'^cos (ad + 2 ■ 7t/2), 

f"'{e) = — a^sin {ad + n) = a^ cos {ad + ^ • 7r/2), 



'• /"(^) = ^" cos (a^ + n • 7r/2). 



80. ^«cA of the successive cle7'ivatives of f (x) equals the 
x-rate of the 2y^'ecedm(/ derivative. 

For /«(.x) = (//" - ' (cc) / cZcc = the x-rate of /« - X^) . 

Cor. f^~^(x) is an increasing or a decreasing fnnction of 
X according as fix) is positive or negative ; and conversely. 

EXAMPLES. 

1. f{x) = cx^ + ax2 + a. /'"(x) = 6 c, />v(x) = 0. 
Here/''(x) changes at the rate of 6 c to 1 of x (§ 80). 

2. f{x) = x« + 4 x4 + 3 X + 2. /vi(.r) = [0. 

3. /(x) = X logx. /"(x) = (- 1)"--'|h-2x'-". 
nere/"-i(x) changes at the rate of (— 1 )"-- [/< — 2 x'~" to 1 of .r. 

4. /(x) = rtx"'. /"(x) = am {>ii — !)(//; — 2) • • • {m — n+ l)x"'-". 

5. /(x) = r' U)g X. /»•(.,•) = X- ». 

7. f{x)= ,\x-'^(logx- 5 /('.). /"(.r) = (-l)«-»[nj--_4xS-». 

V 8. ./•(.)•) = (x- - ;i X + .•>) (■-■'. /-"(x) = Sx'^(^*: 

9. /(x) = .r« k)g X. / v.(.r) = - [4 .r-s. 



60 DIFFEKENTIAL CALCULUS. 

10. f{x) - x^. f'\x) = x^ (1 + log x)2 + x^-1. 



11. y 



x3 #y _ 24 

1 — X ' dx^ ~ (1 — X)5 



12. /(x) = e«^. /"(x) = a" e«^. 

13. f{e) = sin a ^. /"(^) = «« sin (a^ + n ■ 7r/2). 

14. f{x) = (1 + X)"'. /"(x) = m {m - 1) • • • (??i - ?i + 1)(1 + x)"^-« 

1 f7«„ (— l)«4"|7i 

1^- ^-4^T^ = (^- + ^)-^- ^-(I^T^i- 

' ^ 2 +x dx" (2 +x)« + i' 

1^ = - 1 + ^r^— = - 1 + 4 (2 + x)-i. 

2 + X 2 + X ^ ' 

^ 6x-l . d"i/^ -5(-l)''3"[n 

^~3x + 2' dx« (3x + 2)« + i 

^^- y 4x-^-l dxn ^ ^ ^ ^\{2x-l)n + ^ (2x+l)«+iJ' 
^^^^^ EE (2 X -l)-i- (2 x + l)-i. 

Prove each of tlie following differential equations : 



19. When u = Vsec 2 x, dht/dx'^ = 3u^ — u. 
du sec 2 X tan 2 x 



c^x Vsec 2 X 



rt tan 2 x. (1) 



d^u ^ ^ du . ^ „_ 

.-. -r-z = tan 2 X -— + 2u sec^ 2 x. 
dx2 dx 

= u tan2 2 X + 2 M sec2 2 x. by (1) 

= u (sec2 2 X — 1) + 2 M sec2 2 x = 3 it^ — w. 

20. When w = e^ sinx, ^* - 2 ^ + 2 it = 0. 

cZx2 dx 

21. When u = a sin (logx), x^ -— + x - — \- u = 0. 

^ ° ^ dx^ dx 



SUCCESSIVE DERIVATIVES. 61 



22. When y = e'^^ sin mx, ^ - 2 c f^ + (c2 + m2) y = 0. 

23. When ?/ = sin (sinx), — - + -j- tanx + y cos^x = 0. 

^ ' dx^ dx 

24. When u — cos (a sin-ix), (1 — x^) —- — x ^- + cAt = 0. 

^ ^^ '^ dx^ dx 

fl^H flu 

25. When u = (sin-i x)2, (1 - x2) — ^ - x f^ = 2. 

26. When 2/ = ^ (e^/« + e-^/«), ?^ = -^-• 

2 ^ '' (Zx2 a2 



Of each of the following implicit functions obtain that derivative which 
is given at its right : 

27. Iiy'^=2xy — c, d^y/dx^ = c/{x — yY. 

.(,2) = ,(2x,-c);...| = ^^. (1) 



_ d22/_ ^-^ -^ dx -^ \dx ) _ ;;^ dx 

' ' dx2 "~ {y — x)2 ~ (Z/ — a:)2 

From (1), (2), and the given equation, w'e obtain 

d^y _ y^ — 2 x?/ _ c 



(2) 



(Zx2 {y — x)'^ (X — yY 

28. If?/2 = 4px, #?//tZ.t3 = 241)3/^^ 

#/cZx = 2i)/?/; 

d^y _ —2pidy/dx) _ _ 4p^ 

' ' dx2 ?/2 yS 

d^l _ 12 y^p-^dy/dx) _ 24 p^ _ 
dx^ y^ y^ 

29. If .x2 + 2/2 - ,^2^ d-il/dx- = - r-/iY. 

30. If //' r,- (f-x, (l-!//dx~ = — 2 (f< /«) (/'\ 

31. If x2 / ((2 4- ,/J / //.: = 1 , ,/•:// /,/.,.•.: = - 6V a-i/^. 

32. If x2/(r2 - !/-/h~ ~ 1, d-i//dx- = - b*/a-y^ 



62 DIFFEEENTIAL CALCULUS. 



33. ur. = ^^-,'^ = ±^^ 



2a— X dx^ Vx (2 a — x)^ 

34. If x2/3 + ?/2/3 = a2/3, d'^y/dx'^ = a2/3/ o ^1 /3 3^4 /3. 

35. If ?/2 — 2 ax?/ + x2 = c, 

d^ _ (a2 - 1) (y2 - 2 axy + x2) _ c{a'^-l) 
dx^ {y — ax)3 (y — ax)^ 

36. If ?/3 + x3 =: 3 axy, d^y/dx'^ = —2 o?xy / (?/2 — axf. 



37. If e^ + 2/ = x?/, — ^, = — ^^^-^-^^ ■T-'- 77^ ^ 

^' (Zx2 x2(?/— 1)3 



81. Leibnitz's theorem is a formula for the n\h differential 

of the product of two variables. 

Let II and v be functions of x \ then 

d {liv) = du -v + u dv. (1) 

In general, du and c/y will be functions of x ; hence, 

= fZ^zi . V + 2 fZ?i ^z; + w d'^v. (2) 

. • . 6^^ (z^z;) = dhi -v + ^ d^u du -\- 3 du d^v + ic d^v. (3) 

The coefficients and the exponents of operation in (2) and 
(3) follow the laws of the coefficients and exponents in the 
Binomial Theorem. However far we continue the differen- 
tiation, these laws will evidently hold ; hence, we have 

n (n — 1 ) 
d^iuv) = d^u • V + nd^'-^u dv ^ ^—^ — - cZ"~^ u d'^v + • • • 

-\- n du d''~'^v + u d''v. (4) 



ACCELERATION. 63 

EXAMPLES. 

1. rind#(e«^x2). 

Here u = e"^, d»u = a»e''^ dx"^ ; 
and V = x2, dv = 2x dx, d^v = 2 dx^, dH = 0. 

Substituting tliese values in (4), we obtain, when n = 5, 
d^ (e«^' x2) = (a5e«* • x^ + 5 • a^gax . 2 x + 10 • a^e^x . 2) c^x^ 
= ciH^^ (a2x2 + 10 ax + 20) t^x^. 

2. Find d"(x2 sinax). 

Here u = sin ax, (Z«m = a'* sin (ax + ?i • 7r/2) cZx** ; 
and v = x2, du = 2 X (Zx, cZ'^u = 2 (ix^, cZ-^y == 0. 

Substituting these values in (4), we obtain 
d!« (x2 sin ax) /dx" = x'V* sin (ax -\- n • 7t/2) 

+ 2 »2xa'*-i sin [ax + (ji — 1) 7r/2] 

+ 7i{n — l)tt«-2sin[ax + {n — 2)7r/2]. 

3. d" (xe^) = e-'Xx + n) dx^. 

4. (Z" (x2e«a;) /dx'i == (jm-2 e«x [-^^2332 -|- 2 a?zx + ?2 (71 — 1)]. 

5. d» (x^a^) = a^ (log a)»-2 [(x log a + 7i)2 — n] dx'K 

6. d« (x2 log x) = 2 (— 1 )« - 1 |n — 3 x2 - "• cZx". 

82. Acceleration is the time-rate of the velocity v. 
Hence, if s = the distance, and a = the acceleration, 

ds d ds dh 



V = -T-^ a = -r -r = -rr, ' 

dt dt dt dt^ 



(1) 



EXAMPLES. 

1. A point moves along the arc of Ihe parabola y- = A-px with the 
constant velocity u' ; lind its acceloration in the direction of each axis. 
From example 20 of § .'>{), we obtahi 

dx yv' ^^V _ 2;)n' 

dt -Vyi + 4p--i dt ~ Vj/2 + 4^ 

d'^x _ SpH"^ d-y _ _ ip'^yv"^ 

•■• dt'^ ~ (.v2 + 4 ?)-i)--i' dt-i ~ (yJ + 4i.-V- ■ 



64 DIFFERENTIAL CALCULUS. 

The velocities in the directions of the axes are the time-rates of x 
and y in the first quadrant. 

Since (Px/dP- is positive, the velocity in the direction of the x-axis 
continually increases. 

Since for y positive dry / dP- is negative, the velocity in the direc- 
tion of the ?/-axis is constantly decreasing. 

2. Find the accelerations required in example 1, when the path of the 
point is 

(1) an arc of the circle x^ + ?/2 = r^, 

(2) an arc of the elhpse — + tt; = 1, 

x^ ifi 

(3) an arc of the hyperbola ^ — 7-;= 1. 

3. If s denotes the number of feet a body falls in t seconds, and 
g = 32.17, s = gt^/2 is the law of falling bodies in a vacuum near the 
earth's surface ; find the velocity and the acceleration. 

Ans. V = ds/dt = gt; a — dP-s / dP = g, sl constant. 

4. Given s = cP^'^ -^ find v and a at the end of four seconds. 



CHAPTER V. 
INDETERMINATE FORMS. 

83. The value of a function of x for x = a usually means 
the result obtained by substituting a for x in the function. 
When, however, this substitution gives rise to any one of the 
indeterminate forms 

0/0, op I op, 0.(^, ap-ap, 0^ ap\ \^, 

the definition given above is inapplicable and must be enlarged 
as below : 

The value of a function for any particular value of its 
variable is the Ihnit which the function approaches when the 
variable approaches this particular value as its limit. 

This definition is of general application, but it is practically 
useful only when the ordinary and simpler definition fails. 

fix) is often written without the parentheses, as fv. 

The expression /r]„ denotes the value oi fx when x = a. 

EXAMPLES. 
By principles of limits prove that 

When x = a tliis fraction assumes the indeterminate form 0/0. 
Hence, to evaluate it for x = a we must find its limit when x = a. 
For values of .t: other tlian a, we have 

(x — ay^'^ _ (X — aV ' '- _ (.(• — (1)1/1- 

(aj'-2 — a-)!/'* - {X — aY'^'- (x + a)^'^ - \x + ay '^ ' 

limit r (x-a)i/a -| ^ limit r (.r - fl)Wi-2 -i _ __0_ _ ^ 

That^ is, the given I'l-action injuals zero when .r ^ (r. 



66 DLFFEREXTIAL CALCULUS. 



2. 



_a3-| _ 3a_ x5-l -] ^ 5_ 



5. (1 — cos ^) /sin ^] = 0. 

For values of 6* other than zero, we have 

1 — cosg _ 2sm2(g/2) _ e^ 

smO = 2sm(#/2) cos(^/2) "^ ^^2 

limit rl — cos6'l limit f H ^ 



84. If lfx/<f.xl, = 0/0, then lfx/<t>xl =lf'x/<l>'x],. 

That is, if the ratio of two functions of x assumes the form 
0/0 when x = a, /^Ae?i 2^7ie 7'a?^io o/" these functions when x = a 
26- equal to the ratio of their derivatives when x = a. 

By [33] of § 55, the limit of the ratio of the increments of 
two variables is equal to the ratio of their differentials ; hence, 

limit V f{x + ^x) - fx "I fx ■ dx fx 



1 ./^ 



Aa3 = [_(/) {x + Aa?) — ^x_\ cf>'x • dx (fi'x 

Substituting a for x and remembering that /a = c/xx = 0, we 
obtain 

limit r /fa + Ax') "| _ fa ^ ,/^~] _ f'xl 

If f'x/(f>'x^a also assumes the form 0/0, by the principle 
just proved we have 

f'x/cj>'x-]„=f"x/cf."x^^; 

and so on, until we obtain a fraction which does not assume 
the form 0/0 when x = a. 



INDETERMINATE FORMS. 



67 



EXAMPLES. 

1. logx/(x-l)]i = l. 

logx H _ logx ~| _ l/x ~| _ 

x-lji~0' •■• x-lJi~ 1 Ji~^- 

2. (l-cosa;)/x2]o=l/2. 

1 — cos x~\ 1 — cos x~\ sin x~\ 



§84 



x^-l 

x« 

qX — q- 



cos x~\ _^ ^ 1 — COS x ~\ _ sm x~j _ 
^2 Jo~0' •'• x-^ Jo~^^Jo~0 
sin x~| _ cosx ~l _ 1 ^ 1 — COS x ~l _ 1 

["11 . _ /sin7ix\"»~| 

- = - • ^ 9. ( ) = 'nPK 

IA\ n \ X / Jo 

x2 — X ~| _ 

1 — X + logxJi" ^* 



sni X Jo 

gx _ e- a-- — 2 X' 



X — sin X 

g^' — 6- 

X _io 



]. 



^^ xiog(l +x) -[ ^2 
1 — cos X J 



'], 



a 

logT- 
^ b 



-- tan X — sin x~| _ 1 



X — sin-^x ~[ _ _ 1 , 
sin^x Jo 



a* + i — 6^+1- 



x + 



85. PFAe?i 



13. 
14. 



sin^x 
X-'' — X 



1 — X + log X 
sec^x — 2 tan x 



]. 



2. 



c^x — 2 tan x~| _ 1 

1 + cos 4 X J n- / 4 2 



fx~\ ^ ^ an fx~\ f'x'l 

- — assumes the form -^^ ■ — = '— — 

inn A*~l ^ I/.AH 

When '^ " = ' , \ = X- 

limit ri /./>"]_ limit n".v I {fxf \ 



iloiicc 



(1) 



^ 71 



§ 84 



If two variables li;iv(^ ecjual limits, any iN|uinuilt iples of 
those variables have equal limits. 

Hence, multiplying- by (^/.V)-/ (^(/).rV-, \vi^ obtain 

limit 



limit r./''"! 

x = (i Lv>'J 



Lc/>'r_ 



or ^\). 



68 DIFFERENTIAL CALCULUS. 





EXAMPLES. 


1. 


CSC 2x/csc 5x]o = 5/2. 




CSC 2 X sin 5 X , 

r— = -. — ^r~ = t: when x = 0. 

CSC 5 X sm 2 X 




CSC 2 x~j _ sin 5 x~j _ 5 cos 5 x~| _ 5 

CSC 5 x_ ~" sin 2 x_ 2 cos 2 x_ o 2 



§84 

Here we transform the given fraction into an identical fraction 
which assumes the form 0/0 when x = 0. 

2, log x/cscx]o = 0. 

log x-| _ -ap _ logx -| _ 1/x -j . 

]_ — sin^x"! _ _ 
xcosxJo 
^~1 _ — 2 sin X cos x "l _ 
X J cos X — X sin xj o 



cscxJo ap cscxJo — esc x cot 

1/x _ — sin% _ 1/x 

CSC X cot X " X COS X ' " " — CSC X cot 

— sin^x"! — 2 sin x cos x 



X cos 



84 



Here we derive a new fraction by § 85, and transform that into an 
identical fraction which assumes the form 0/0 when x = 0. 



q =0, 6. -5?5f1 = 

:Jo sec oxj 77/2 



logx~| ^ ^ sec; 

cotx 



4 tanx -[ ^^ ^ iog(l - x) -j ^^ 

tan3xj7r/2 ' ' sec(7rx/2)Ji 



^ log(x-;r/2) -| ^^ ^ log tan 2 x H ^ 

tanx Att/2 ' ' log tan X J 77/2 

1 =p-^^ct5^y-| =1X1 = 1. 

Jo L ic V X / Jo 



(e^ — l)tan^x ~1 _ f e^ — 1 /tanx^2 



_ tan X — x ~| _ 11"'^ — ^^^^ ^ "'" ^^^ ^"1 — 

' X — sinxJo ' ' sinx + cosx — lJ,r/2 



86. The forms • ap and ap — ap. A function of x which 
assumes the form • ap ov ap — ap when cc = a is evaluated by 
first transforming it into an identical fraction which will 
assume the form 0/0 or ap / ap when x = a. 



INDETERMINATE FORMS. 

EXAMPLES. 
1. (1 —x) tan (;rx/2)]i = 2/ TT. 

Taking the reciprocal of the infinite factor, we have 

(1 — x) tan — - = — — -— = - when x = 1. 

^ ' 2 cot(;rx/2) 

2. sec 3x cos 7x]77-/2 ^^ 7/3. 5. sinx logx]o = 0. 

3. (l-tanx)sec2x];r/4 = l. ^- a^log^]o = 0. 

^ r 2 1 "I _ 1 

4. tanxlogsinx];r/2 = 0. 7. |^^2 _ i ~^^^Ji~ ~2 

2 1 1 -X ^ 

when X = 1. 



x"^ — 1 X — 1 x2 — 1 



8. . = — 1. 10. X tan x — - sec x = — 1. 

LlogX logxJi L 2 J;,/2 

Lsm^x 1 — cosxJo 2 ^\ x/Jx 

Limit r , /^ , a\~| log(l+«2;)~l , 1 

^^^Lxlog(l + -)J=^^ 'Jo=^'^-^-^^ = x- 



87. The forms 0°, ap\ and l^'^. When for .t = «, a func- 
tion of X assumes one of the forms 0*^, c^°, or 1-^, the loga- 
rithm of the function will assume the form ±^ ■ ap^ and can 
be evaluated by § 86. From its logarithm the value of the 
given function can be obtained. 

EXAMPLES. 
1. x^-]o = l. 

log(x') = x log x — —^'ip when x = 0. 

X log x]o = 0. § 80, example 

.-. logx-']o = 0; .-. x']o = 1. 

log ( 1 + - )' 1 = -P l^\ii- ( ^ + " )] — '^- § ^^*' example 1 1 



70 DITEERENTIAL CALCULUS. 

3. x^in^Jo = 1. 8. (1 + ax)i/^]o = e«. 

4. (sin X) tan .T]^^ 2 = 1. 9. (logx)-^-]o = 1. 

5. xi/a-^)]i = e-i. 10. (e^- + x)i/^]o = e2. 

6. (cos mx) n /x]o = 1. 11. (cos 2 x)i /•'-]o = e-2. 

7. xe"-i]o = l. 12. (logx)^-i]i = l. 

88. Evaluation of derivatives of implicit functions. 

When y is an implicit function of x, its derivative, though 
containing both x and y, is a function of x. Hence, when the 
derivative assumes an indeterminate form for particular values 
of X and y, it can be evaluated by the previous methods. 

EXAMPLES. 

1. Find the slope of a-y'^ — a-x- — x^ = at (0, 0). 

Here ~f- = ——, = - ? when x = y = 0. 

dx 2a^ 

dy-l _ 2 cfix + 4 xn _ 2 cfi + 12 x^ n _ 1 n 

^^ ^' dxj 0,0 ~" 2 a2?/ J 0,0 ~ 2a- • cZy/dxJ o,o "" dy/dxj o,o ' 
.-. (d?//cZx)2]o,o = 1, or d?//dx]o,o = ± 1. 

2. Fmd the slope of y^ = ax^ — x^ at (0, 0). 

^ d?/-| _ 2 CTX - 3 x2 -| _ 2a — 6x H 
^^^dxJo,o~ 3?/2 J 0,0 6?/-d?//dxJo,o' 
/^yn _ 2 a — 6 x "| _2a_ , ^"] _ , 

"\(^x/ Jo,o~ 6?/ Jo,o~ ~^' °^ tZxJo,o"~ ^" 

3. Fmd the slope of x^ — 3 axy + 2/3 = o at (0, 0). 

An%. (i?//(ix]o,o = or c^. 

4. Fmd the slope of x* — a^x?/ + 6"-?/2 = at (0, 0). 

Ans. cZ?//dx]o,o = or a2/62, 

5. Find the slope of (?/2 + x2)2 — 6 ax2/2 — 2 ax^ + a2x2 = at (0, 0) 
and (a, 0). ^^s. d2//dx]o,o = ± o^ ; dy/dx^a^o = ± 1/2. 



* CHAPTEK VI. 

EXPANSION OF FUNCTIONS. 

89. A series is a succession of terms whose values are all 
determined by any one law. A series is finite or infinite 
according as the number of its terms is limited or unlimited. 

The sum of a finite series is the sum of all its terms. 

The sum of an infinite series is the limit of the sum of its 
first n terms as n increases indefinitely. When such a limit 
exists, the series is said to be convergent ; when no such limit 
exists, the series is divergent and has no sum. 

For example, the series 

1 + 1/2 + 1/4 + 1/8 4- • • • + l/2"-i+ • • • 

is convergent ; for tlie sum of its first n terms = 2 when n = qo. 
The series 

1 - 1 + 1 -1 + • ■ • + (-l)"-i + • • • 

is divergent ; for the sum of its first n terms does not approach a Uniit 
wlien w = CO. « 

90. To expand a function is to find a series tlie sum of 
which shall equal the function. Hence, the exjxnisiou of a 
function is either n finite or a convergent infinite series. 

When the expansion is an infinite series, the diiforence 
between the function and the sum of the first // terms of tlie 
series is called the remainder after n terms. AVlien // = oc. 
this remainder must evidently approach zero as its limit. 

For example, by division we obtain 

-r^ ^ 1 + X, + .c- + x^ + • • • + .(•" 1 + , ^— • (1) 

1 — a; 1 — .r. ^ ' 



72 DIFFERENTIAL CALCULUS. 

Here x'V(l — ^) is the remainder after n terms. When n = oo and 
X >> — 1 and <; 1, this remainder = 0, and therefore the sum of n terms 
of the series = tlie function ; but when n = ^ and x > 1 or <; — 1, the 
remainder increases arithmetically, and therefore the sum of n terms of 
the series diverges more and more from the value of the function. 

Hence, the series in (1) is convergent and its sum equals the function 
only for values of x between — 1 and + 1. 

Some functions may be expanded by division, as above ; 
some by indeterminate coefficients ; others by the binomial 
theorem ; and so on. The binomial theorem, the logarithmic 
series, the exponential series, etc., are all particular cases of 
Taylor^s theorem, which is stated and proved in § 92. 

For the proof of this theorem we need the following lemma : 



91. Lemma. If (j>z a7id cji'z are each continuous hetiveen a 
and a + h, and cf)a = (fi (a -}- h) = ; (fi'z must equal zero for at 
least one value of z between a and a + li ; that is, <f>'(a-\- 6h) = 0, 
where 6 is some positive proper fraction. 

For if <j>z is continuous and <^a = ^(a-\-h) = 0\ then, as z 
changes from a to a -\- h, cfiz must first increase and then 
decrease, or first decrease and then increase ; hence, (f)'z must 
change from + to — or from — to + ; and therefore, if con- 
tinuouSj it must pass through for some value of z between 
a and a + h. Denoting this value of zhj a -\- 6h where has 
some value between and + 1, we have <j>' (a + 6h) = 0. 

92. Taylor's theorem. When fz, f'z, f"z, • • ;f''z are each 
continuous between x and x -\- h, 

f{x + h) = 

where the last term is the remainder after n terms, and is some 
positive proper fraction. 



TAYLOR'S THEOREM. 73 

Let Ph''/\n denote the remainder after n terms when x = a] 
then we have 



f{a + h) 

\1 






We proceed to find the value of P. 

Putting h = h — a\xv (1), and transposing, we obtain 

J * 1^-1 ^ \n -^- ^^> 

Let (f>z represent the function of z obtained by substituting 
z for a in the first member of (2) ; then 



^z=fb-fz-fz-—--f"z 



9 



Differentiating (3) to obtain cj^'z, we find tliat the terms of 
the second member destroy each other in pairs with the excep- 
tion of the last two, and obtain 

\>i — 1 \'?i —1 ^ ^ 

By hypothesis /:;, fz, • • •, f''z are continuous between a and 
a + li\ hence, from (3) and (4), it follows that f^z and <^',: are 
continuous between a and a + h. 

Putting a for z in (3), by (2) wo liavo <^a 0. 

Putting h for ,-:; in (3), we have ^h _ 0, I.e. <f) {^d + h) = 0. 

Hence, by § 91 we have 

cf>'(a -hO/i) -0, (5) 

where 6 denotes some positive proper fraetion. 



74 DIFFERENTIAL CALCULUS. 

Putting a + Oh for z in (4), by (5) we obtain 

Substituting this value of P in (1), and then putting x for 
a, since a is any value of x, we obtain (A). 

Formula (A), called Taylor's theorem, was first published 
by Brook Taylor in 1715. 

CoR. 1. Putting a? == in (A) and then substituting x for 
h, we obtain 

/x=/0+/'0~+/"Op + ...+/»-'0||^+/»(fe)^- (B) 

Pormula (B), called Stirling's or Maclaurin's theorem, was 
first given by James Stirling in 1717. It is a special case of 
Taylor's theorem. 

Cor. 2* Denoting the remainder after n terms in (A) by 
Hj^, and in (B) by i?j^, we have 

i?^=/"(x + e/0^> i?,,=/"(to)|- (6) 

The form of Bj, in (6) is due to Lagrange. 

CoR. 3. If i?y = when ?i = oc, the series in (A) is the 
expansion of f(x + h). If Iijj =^ when n = co, the series in 
(B) is the expansion of fx. 

CoR. 4. If f"(x) increases (or decreases) from /''(x) to 
/" (x 4- h), and we take the sum of the first 7i terms in (A) 
as the value of f(x + h), the erro7' lies between 
f (x) • h"/\n and f" (x + h) IV' /\n. 

Jlf'ix) increases (or decreases) from /"(O) to /""(x), and 
we take the sum of the first n terms in (B) as the value of 
f{x), the error lies between 

f^(())x''/\n and/«(cf)ccY|^. 

* Cors. 2, 3, 4, §§ 93, 96, and the proofs for convergency in §§ 94-98 
may be omitted in the first reading of this chapter. 



EXPANSION OF SIN X. 75 



/y>'* rv» rY" ■O^ 

93. Since 7- = 



n 12 3 yi — 2 71 — 1 yi 

x^ /\n = wlien £c is finite and y^ = go. 

Cor. When ?i = go and a; is finite, 

J?2. = when /^(cu + Oh) is finite, 
and Rjij d= when /" {9x) is finite. 

94. To expand sin x a?ic? cos x. 

Sin 0? is a particular case of fx ; hence, to expand sin x we 
use Maclaurin's theorem, or (B). 

Here fx = sin x, .\fO = 

f'x = cosx, .'.f'0 = l 

f"x = -smx, .•./"0 = 

f"x=-Gosx, .•./'"0--1. 

Since /'^^ = sin x —fx, 

the four values given above will recur in sets. 
Substituting these values in (B), we obtain 

\o IT) . I I \2n — 1 ^ • 

The 7it]i term in (1) is readily written out by ius[)ertic)n. 

Proof of (1).* Since /".x- = sin (.r + ;/ • 7r/2), fx ami all its 
successive derivatives are continuous and finite lor all values 
of X. Hence, by Cor. of § 9o and Cor. ,'> of § 92, the series m 
(1) is the expansion of sin x for all liuite values of x. 

* For a discussion of ooiivci-uiMit scrii>s t-onsuli Tax lor's *• Collogo 
Algebra," Osgood's " Introdurliou U> Intiniic Si>rics/' or s.muc nunv 
extiMided work on the Calculus. Vov the lu'oot' of ctui\crucncy the scope 
of (liis work limits us to tlu> use of the rcutaiudiu- after n terms in (^A) 
or (B). 



76 DIFFERENTIAL CALCULUS. 



In like manner we obtain and prove 

li |6 + '"+| 2(n-l) 



cos^-l-T^+^-T77+--+ |^^,^^,, +'-. (2) 



Identity (2) could be obtained by differentiating (1). 

95. To deduce the exponential series. 

^, _ x\oga (x log ay (x log ay-'' 

^-^+ 1 "^ |2 +*••+ 1 ^-1 + • "• ^^) 

Froof of (1). Here f"x = (log a)" a*. 

Hence, when « is positive, /x and all its successive deriva- 
tives are continuous for all values of x. 

When x is finite, (X^^ is finite. 

By § 93, (x log <x)'Y|?z = when n = oo and a? log a is finite. 
Hence, i^j^ = when n = cc and cc is finite. 
Therefore, the series in (1) is the expansion of a^ when a is 
positive and x is finite. 

CoE. 1. Value of e^ Putting <x = e in (1), we obtain 

Cor. 2. Value of e. Putting a? = 1 in (2), we obtain 

-l + l + | + | + 4+--- + |^ + ---- (3) 
= 2.718281- • •. 

96. Second form of remainder. If we denote the remainder 

after 7i terms in (1) of § 92 by Fih, and proceed as before, we 
obtain 



LOGARITHMIC SERIES. 77 

97. To deduce the logarithmic series. 

log«(l+x) = m(^x-- + - + ^— ^^ + -.-| (1) 

Froof of (1). /"x = (- \y-^\n_-^l {\ + ;z;)'\ 
When £c < — 1, log (1 + .x) has no real value. 
When £c = — 1, the odd derivatives are discontinuous. 
When a:> — 1, /ic and all its successive derivatives are con- 
tinuous. 

Using the second form of R^^^ we obtain 



X^QxJ 1-0 

The second factor in i?^^^ is finite, and the first factor = 
when 71 = CO and x'> — 1 and < 1 or cc = 1. 

Therefore, the series in (1) is the expansion of log (1 + x) 
when X >» — 1 and < 4- 1 or x = 1. 

Putting X = 1 in (1), we obtain log„2. 

Putting ir = 1/2, we obtain log„ (3/2), or log„3 — log„2; etc. 

Cor. 1. To deduce a series more rapidly convergent than 
that in (1), we put — x for x in (1) and obtain 

log, (1 - x) = VI ( - ^ - 2" ~ ^ ~ 4" "^ ' ■ / ^'^^ 

Subtracting (2) from (1), avo obtain 



1 + .^; ^ / . .* 



1-x 



>.3 



X . X 



lo^- TZr:' = 2 m X + - + '.+- + •••• (3> 



) i) i 



Let .^ = ^_-_^^; then J-— ^ = -— -. ^.n 

Substituting in i^^\) the vahics in (^h, wt^ cUUain 



78 DIFFERENTIAL CALCULUS. 

/. log, (^ + 1) - log„^ + 2 m(^^^^ + ^^^^^^ + • • • J. (6) 

When ^>0, 0<a:<l; hence, the series in (6) is conver- 
gent for all positive values of z. 

Log,(^ + 1) can be readily computed when log,^ is known. 

CoR. 2. If m = 1, <2 = e and (5) becomes 



^2^+1 3(2^ + 1)^ 



log^ = 2(,,^^ + ,,,^ , ..3 + ---)- (7) 



Dividing (5) by (7) and denoting {z + V) j z by iV, we have 
log^iV^/ log N = m, or loga N = m log N. (8) 

Cor. 3. Value of m. Putting iV^= a in (8), we obtain 

1 = m log a, or m = I /log a. (9) 

CoR. 4. Value of M. If M denotes the modulus of the 
common system whose base is 10, from (9) we have 

^^ = r^ = oQHo-Qrc = 0.434294- • •. 

log 10 2.302o85 

98. To deduce the binomial theorem, or 

(X + hy = X"' + ma^"'-' h + ^^H^^^^~ ) . ^m-2 J^2 

7n(7)i ^ l)(ni — 2) Q70 , 

H ^ -^ ^ x'''-^h^ + • • • 

H !^ ^- ^ ■ — ^a:'"-" + Vz."-i + • • •. (1) 

|?z — 1 

(:r + /?)'" is a particular case of /(ic + h); hence, to expand 
(x -\- hy we use Taylor's theorem, or (A). 



BINOMIAL THEOREM. 79 

Here f{x + A) = {x ^- A)'" ; .'.fx = x^, 

/'a? = mx"*~^, /"a? = 711 (in — 1) ^"'~^, 

/'"x = m (771 - 1) (m - 2) x'"-^ 

yn-i^ = ^n(7n -!)••• (m -?i + 2)£c'"-" + i. 
Substituting these values in (A), we obtain (1). 

Froof of (1). /"cc = 771 (jn — !)••• (7n — 7^ + 1) cc"*"**. 
Hence, fx and all its successive derivatives are continuous 
for all values of x. 

Using the second form of Ej,, we obtain 

771 (m — !)• • -(m — 71 + 1) _ fh — Oh\" (x + (9/i)'" 



^^ [7^-_i \x + ^Ay 1 - ^ 

When X > A arithmetically, and n = oo, the product of tlie 
first and second factors = ; hence, Ej, = 0. 

Hence, the binomial theorem holds true when the first 
term is greater than the second arithmetically. 

When m is a positive integer, we obtain 
f"^ + 'x = 0,f" + ^^x = 0, •••; 
hence, in this case, the expansion in (1) is a finite series of 
m + 1 terms. 

99. Failure of Maclaurin's theorem. Tlie successive 
derivatives of log x are ap and discontinuous wlien a* = ; 
hence, (B) fails to expand log x. 

For a like reason, (B) fails to expand 

cot ic, CSC .r, vers~'.T, a^^^, sin(l/.7'), • • •. 

When (B) fails to expand fx, (A) will fail to expand 
f(x 4- A) for X = 0. 

For cxamplo, whon .r ~ 0, (A) fails (o expand \oiX{x -f //), cot (.r + ?i). 
vers-i(:c + A), • • •. 

(A) may fail to oxpaiid f(x + //) ior other valines than .r 0. 
'^riie limits botwiHMi which any (Expansion lu>hls true shouKl 
be carefully determiniHl. 



80 DIFFERENTIAL CALCULUS. 



EXAMPLES. 



1, tan X = X + — + -— - + -— — + • 
o 15 olo 



„ ^ . X'^ , X» X' x^ 

2. tan-i X = x — ^ + — — — + — — •••• 

Here /'x = (1 + x'^)- 1 = 1 - x2 + x* - x^ + 
.-. /"x = — 2 X + 4 x^ — 6 x5 4- • • •, etc. 



^ . , , 1 x3 , 1 • 3 x5 , 1 • 3 • 5 x^ ^ 

3. sm-ix = x + --- + — •- + ^:-^.y + 



4. From the series in example 3 find the value of tt. 
Putting X = 1 /2, we obtain 

.-. IT = 3.141592- • •. 



, , ,x2 5x» 61x». 

5. secx=l+- + ~ + -^ + 



, X2 3X4 8j,5 3x6 



7. 


X^ X^ X* 

log (1 + sin «)=a:--+-- — +•••. 




8. 


' / h^ h^ h^ 
sin (X + ;i) = sinx (^ 1 - — - + 1^ - y- + • 


■■) 




+ eosx(^-- + ----+. 


■■) 




= sin X cos /i + cos x sin /i. 




9. 


cos (x + h) = cos xcosh — sin x sin h. 





§94 



10. 10ga(X + ;i) 

^ \x 2x2 3x3 yix"- / 



EXPANSION OF FUNCTIONS. 81 

11. g... = a.>(i + ^Al^g^ + JA}^ + ■ . . + i^l^^g^V;-^ + ...). 

12. log(l+e-) = log2 + ^ + |-|^+---. 

13. Itfx =/(— x), the expansion olfz will contain only even powers 
of X ; while if/x = — /(— x), the expansion of /x will involve only odd 
powers of x. 

14. What powers of x will appear in the expansion of sin x ? cosx ? 
tanx? secx? sin-ix? tan-^x? (e^ + l)/(e^ — 1) ? 

15. Regarding h as an increment of x, find from (A) the value of the 
corresponding increment of /x, h being reckoned (1) from any value of x, 
(2) from X = 0. 

Compare the second result with (B). 

16. Prove geometrically that 

f{x + h)=fx + hr{x + eh), 

fx and/'x being continuous between x and x + h. 

17. When x = 2, prove that 

X — sin X = V'i0 ; 1 — cos x = v^i^ ; § 04 

a"" — 1 = Vii ; X — tan x = i^si^ ; § 95 

X — tan— 1 X = Usi^ ; ] — sec x = Uoi"^- 

18. From (2) of § 95 and (1) and (2) of § 94, show that 

e^ "^ = cos X + V— 1 sin x, 
and e~^ ~^ = cos x — V— l sin x. 

19. From the results in example 18, obtain tlie cxponcntuxl values 

sui X = 7 > cos X = 

2 V — I ^ 

20. Find the limits of the error when we take (he sum of n terms of 
the series as the value of a-'"; log„(l + .r); (.r + //)'". 



CHAPTER VII. 
MAXIMA AND MINIMA. 

100. A maximum of fx is a value of fx which is greater 
than those immediately preceding and immediately following. 
A minimum of fx is a value which is less than those imme- 
diately preceding and following. In defining and discussing 
maxima and minima of fx, it is assumed that x increases con- 
tinuously, and that fx is a continuous one-valued function. 

101. fx is positive immediately before and negative imme- 
diately after a maximum ofix. ; also fx is negative immediately 
before and positive immediately after a minimum o/fx. 

For fx is an i7ic7xasing function immediately before and a 
decreasing function immediately after a maximum ; also, fx is 
a decreasing function immediately before and an increasing 
function immediately after a minimum (§ 12, Cok. 1). 

102. Any value of x which renders fx a maximum or a 
mhiimum is a root of f x = or f x = op. 

From § 101 it follows that /'a; changes its quality when/x 
passes through either a maximum or a minimum. 

When fx is continuous for all values of x, fx must pass 
through zero to change its quality. 

When fx is a fraction whose denominator becomes zero 
for some finite value of x, fx may change its quality by 
becoming ap. 

For example, when cc — 2 = 0orx = 2, the fraction a/ {x — 2) becomes 
ap and changes its quality. Again, tan x, or sin x/cos x, becomes ap and 
changes its quality when cos x = 0, or x = 7r/2. 



MAXIMA AND MINIMA. 



83 



The converse of this theorem is not true ; that is, any root 
oifx = or f'x = c^ does not necessarily render fx either a 
maximum or a minimum. These roots are simply the critical 
values of x, for each of which the function is to be examined. 

103. Geometric illustration. 

Let adefh be the locus oi y = fx. 

Then fx will be represented by the ordinate of the point (ic, y), and/'x 
by the slope of the locus at the point (x, y). 

By definition, the ordinates Oa., Cc, and Ee represent maxima of fx ; 
while Bb, Dd, and Ff repre- 
sent minima (§ 100). 

The slope f'x is positive 
immediately before a maxi- 
mum ordinate, and neg- 
ative immediately after; 
while the slope is negative 
immediately before a mini- 
mum ordinate and positive 
after (§ 101). 

The slope f'x is or ap at any point whose ordinate fx is either a 
maximum or a minimum. The slope f'x is discontuuwus at the points 
e and/, where it changes its quality by becoming ap (§ 102), 

The slope f'x is at g, and ap ?it h; but it does not change its quality 
at either point, and neither Gg nor Hh is a maximum or a minimum 
ordinate. 

Note. If the points e and / were shooting x'x^ints (§ ir»o) instead of 
cusps (§ 151), /'x would change abruptly from a positive tinite value to a 
negative value, or vice versa; hence, § 102 is not strictly true when the 
locus of ?/ =/(.'•) has a shooting point. 




104. Maxima and minima occur alternately. For botwoon 
two maxima a function must change from a (hn'rcasiitix to an 
increasing function, and hence pass througli a minimum. For 
a like reason, between two minima a function passes through 
a maximum. 

This princijde is evident also from tlio Umuis in § UK>. 



84 DIFFERENTIAL CALCULUS. 

105. Let 3i, be a critical value obtained from f'x = o, and let 
a be substituted for x in the successive derivatives of ix. 

If the first derivative ivhich does not vanish is of an even 
order, fa is a maximum or a minimum of fx according as this 
derivative is negative or positive. 

If the first derivative which does not vanish is of an odd 
order, fa is neither a inaximum nor a minimum of fx. 

Since fa e: 0, from Taylor's theorem we liave 

f{a-h)-fa=ra.hy2-r^a-h'/\^+r^a.hy\4^-";(l) 
f{a + h) -fa=fa -W J^ +f^^a • hy\Z+f''a • hy\^+-': (2) 

If h be taken very small, the quality of the second member 
of (1) or (2) will be that of its first term ; hence, 

If f^a is -, fa >f(a - h) and fa >f(a + h) ; 

that is, fa is a maximum of fx. § 100 

If f^a is +, fa <f(a - h) and fa <f{a + h) ; 

that is, fa is a minimum of fx. § 100 

If /"<z = and /'"a is not zero, fa is evidently neither a 
maximum nor a minimum oifx. 

If /"a =f^'a = 0, fa will evidently be a maximum or a 
minimum according Sisf'^a is — or + ; and so on. 

Ex. Examine 4 x^ — 15 x^ + 12 x — 1 for maxima and minima. 
Here f'x = 12 cc2 - 30 x + 12, 

and f'x = 24 X — 30. 

The roots of f'x = 12 x^ — 30 x + 12 = are 2 and 1 /2 ; hence, the 
only critical values of x are 2 and 1/2. 

f"{2)= + 18; .•.f{2), or — 5, is a minimum of/x; 
/''(l/2)= — 18; .•./(1/2), or 7/4, is a maximum of /x. 

Note. The student should illustrate these properties of this function 
by constructing the locus of?/ = 4x^ — 15x2+ i2x — 1. 



MAXIMA AND MINIMA. 85 

106. Let ?i he a critical value given by either f'x — or 
f 'x = ap, and let h. he a very small 'positive number ; then 

IfV(d, — h) is positive and f'(a + h) is negative, 

fa is a maximuni of f x. § 101 

7/'f'(a — h) is negative and f'(a H- h) is positive, 

fa is a minimum of f x. § 101 

i/* f ' (a — h) and V (a H- h) have the same quality, 

fa is neither a maximum nor a minimum of f x. 

107. Auxiliary principles. By the following obvious prin- 
ciples we may often simplify the solution of problems in 
maxima and minima : 

(i) Since fx and log (/x) increase and decrease together, any value of 
X which renders fx a maximum or a minimum renders log {fx) a maxi- 
mum or a minimum ; and conversely. 

(ii) Since when fx increases its reciprocal decreases, any value of x 
which renders fx a maximum or a minimum renders its reciprocal a 
minimum. or a maximum. 

(iii) Any value of x which renders c{fx)^ c being positive, a maxi- 
mum or a minimum renders fx a maximum or a mininmm ; and con- 
versely. If c is negative and /a is a maximum, c{fa) is a minimum. 

(iv) Any value of x which renders c -\-fx a maxinmm or a mininuim 
renders /x a maximum or a minimum ; and conversely. 

(v) Any value of x which renders fx positive, and a maxinmm or a 
mininmm, renders (/x)" a maximum or a mininmm, n being any positive 
whole number. 

EXAMPLES. 
Examine /x for maxima and minima when 

1. fx = x^ — 9 x-^ + IT) X — r>. 

/''(I) is — ; .-./(I), or -4, is a mnxiimim oi fx. 
/''(5) is + ; .-./(o), or — 28, is a miniiiuun o( fx. 

2. fx-x^-[>x^ + r).r5-l. 

An^. /(I), or 0, is a max.; /(.">), lU- — 28, is a min. 



86 DIFEERENTIAL CALCULUS. 

3. fx = ^x^- 125 x3 + 2160 x. 

Ans. /(— 4) and/(3) are max.; /{— 3) and/(4) are min. 

4. /x = x3-3x2 + 3x + 7. 

Here f (1) = 0, f (1) = 0, and /'" (1) = 6 ; hence, / (1) is neither 
a maximum nor a minimum of fx (§ 105). 

5. /x = 2x3-21x2 + 36x-20. 

6. /x = x3-3x2 + 6x + 7. 

Ans. No real value of x renders /(x) a max. or a min. 



7. Examine c + V4 a^x^ — 2 ax^ for maxima and minima. 



By § 107 any value of x which renders c + V4 a-x"^ — 2 ax^ a 
maximum or a minimum renders 



V4 a2x2 — 2 ax=^ 4 a2x2 — 2 ax^, or 2 ax^ — x^ 
a maximum or a minimum. 
Hence, we let/x = 2 ax^ — x^, etc. 

Ans. c is a min. ; and c + 8 a- V3 /9 is a max. 

8. Examine h -\- c{x — aY'^ for maxima and minima. 

Let fx = {x — a)2. Ans. 6 is a min. 

9. Examine (x — 1)^ (x + 2)^ for maxima and minima. 

/'x = (X - 1)3 (X + 2)2 (7 X + 5). 
Hence, the critical values are 1, — 2, and — 5/7. 
/' (1 - h) is -, and/' (1 + h) is + ; 

.-./(l), or 0, is a min. of /x. § 106 

/'(- 5/7-70 is +, and/' (-5/7 + /z) is -; 

.-./(- 5/7) is a max. of /x. 

/' (- 2 - /i) and/' (- 2 + Ti) are both + ; 

hence, /(— 2) is neither a maximum nor a miniumm of /x. 

La this example the method in § 106 is preferable to that in 
§105. 



MAXIMA AND MINIMA. 87 

10. Examine -r~ for maxima and minima. 

a — 2x 

f'x = (a - x)2 (4 X - a) / (a - 2 x)2. 

f'x = gives X = a or a/4 ; 

f'x = ap gives (a — 2 x)^ = 0, or x = a/2. § 102 

/'(a/4 — h) is — , and/' (a/4 + h) is + ; 

•'•/(o^/4) is a min. offx. 

When X = a, or a/2, f'x does not change its quality; hence, 
neither /a nor /(a/ 2) is a maximum or a minimum oifx. 

3.2 — 7 ^ _|_ (5 

11. Examine —: — for maxima and minima. 

X — 10 



Ans. /(4) is a max.; /(16) a min. 

^ , for maxim? 
(X - o)'- 



(x + 2)3 
12. Examine ^ rf-. for maxima and minima. 



Ans. /(3) is a max ; /(13) a min. 

13. If /x = x(x + a)2(a — x)"^, /(— a) and /(a/3) are maxima, and 
/(— a/2) is a minimum. 

14. V 2 is a maximum of sin 6 + cos 6. 

15. e is a minimum of x/logx. 

16. 1 is a maximum of 2 tan 6 — tan^ 0. 

17. e^ /<^ is a maximum of x^ z^. 

18. 3 V3/4 is a maximum of sin 0{l + cos 6). 

19. ^/ (1 + tan 6) is a maxinuim when = cos 6. 

Examine the reciprocal of this functic^n for maxima and niinima, 

20. Sliow that 2 is a maxinuim ordinate and — 2(5 a uiininuun onli- 
nate of the curve y = x^ — dx'* + bx^ + I. 

21. Show that a/3 is a niaxinuini ordinate and a uiininuun ordinate 
of the curve // = (2x — ay ' -^ {x — a)- ' '^ . 

22. Sliow that ;>\^;'./l*' i^ :v niaxinnini onlinate of (he curve 

tj '— sin''\r cos .I". 



88 



DIFFERENTIAL CALCULUS. 



PROBLEMS IN MAXIMA AND MINIMA. 

1. Find the altitude of the maximum cylinder that can be inscribed 
in a given right cone. 

Let DAB be a section through the axis of the 
cone and the inscribed cylinder. 
Let a = DC, b = AC, y = MC, x = IM, and 
V = the volume of the cylinder ; 




then 



Ttxy^ 



From the similar triangles ABC and IBH^ 

y = {h/a){a — x); 

AM C KB .-. F = ;r(&/a)2x(a — x)2. 

V will be a maximum vi^hen ic (a — x)^ is a maximum. 
Hence, let fx = x{a — x)^, etc. 

Ans. The altitude of the cylinder =1/3 that of the cone. 

2. Find the altitude of the maximum cone that can be inscribed in a 
sphere whose radius is r. 

Let A CD and A CB be the semicircle and the triangle which gener- 
ate the sphere and the cone, respectively. 
Let X = AB, y = BC, and 

V = the volume of the cone ; 
then V = ^Ttxy'^. 

y^ = AB ■ BI) = x {2 r — x); 
.-. V=i7rx'^{2r — x). 

V will be a maximum when x^ (2 r — x) is a maximum. 
Ans. The altitude of the cone =4/3 the radius of the sphere. 




3. Find the altitude of the maximum cylinder that can be inscribed in 
a sphere whose radius is r. 

X = AB, and y = BE ; 



Let 
then 



B 



V = 2 Ttxy^ — 2'itx (r- — x^). 
Ans. Altitude = 2 r V3/3. 



4. The capacity of a closed cylindrical vessel is c ; find the ratio of its 
altitude to the diameter of its base when its entire inner surface is a 
minimum ; find its altitude. 



MAXIMA AND MINIMA. 89 

Let u equal the radius of the base, x the altitude, and S the entire 
inner surface ; then 

C = TTXM^, (1) 

and S = 2 Ttu^ + 2 tcxu. (2) 

From (1), du/dx = — u/2x. (3) 

From (2), dS = 4 Ttudu + 2 Ttxdii + 2 nudx. (4) 
"When iS is a minimum, dS /dx = 0, or dS = ; hence, 

dht/dx — — u/ {2u + x). (5) 
From (3) and (5), we obtain 

2x = 2u + x, orx = 2u. (6) 

Hence, as S evidently has a minimum value, it is a minimum when 
the altitude of the cylinder is equal to the diameter of its base. 
From (1) and (6), we find the altitude 

x = 2^c/2 7t. 

5. Find the maximum rectangle which can be inscribed in the ellipse 

a;2/a2 +2/2/5-2 = 1. (1) 

Let (cc, y) be the vertex of the rectangle in the first quadrant, and 
let u denote the area ; then 

U = 4X7/. (2) 

Differentiating (1) and (2), and proceeding as in example 4, we 
find that the maximum area is 2 ab. 

6. Find the maximum cylinder which can be inscribed in an oblate 
spheroid whose semi-axes are a and b. 

The ellipse which generates the spheroid is 

xya'^-hyyb-2 = l. (1) 

Let (;c, y) be the vertex in the first (luadrant of the rectangle 
which generates tlie inscribed cylinder ; then 

V=2 Ttf/x-. (2) 

7. The capacity of a cylindrical vi-sscl with open top being constant, 
what is the ratio of its altitude to the riulius of its base w hen its inner 
surface is a minimum '? 

8. A square piece of sheet le;nl has a square cut o\\{ at each corner; 
find the side of the squari> cut out when thc^ remainder of tlte sheet will 
form a vessel of maximum capacity. 



90 DIFFERENTIAL CALCULUS. 

9. The radius of a circular piece of paper is r ; find the arc of the 
sector which must be cut from it that the remaining sector may form 
the convex surface of a cone of maximum volume. 

Ans. Arc = 2 nr (1 — V6/3). 

Let X = the altitude of the cone ; 

then F= 7rx(r2 — x2)/3. 

10. A person, being in a boat 3 miles from the nearest, point of the 
beach, wishes to reach in the shortest time a place 5 miles from that 
point along the shore ; supposing he can walk 5 miles an hour, but row 
only at the rate of 4 miles an hour, required the place where he must 
1^^^- Ans. 1 mile from the place to be reached. 

11. Find the maximum right cone that can be inscribed in a given 
right cone, the vertex of the required cone being at the centre of the 
base of the given cone. ^ns. The ratio of their altitudes is 1 : 3. 

12. A Norman window consists of a rectangle surmounted by a semi- 
circle. Given the perimeter, required the height and the breadth of the 
window when the quantity of light admitted is a maximum. 

Ans. The radius of the semicircle =: the height of the rectangle. 

13. Prove that, of all circular sectors having the same perimeter c, the 
sector of maximum area is that in which the circular arc is double the 
radius. 

Let X = the radius of the sector ; 

X (c — 2 x) 

then area = • 

A 

14. Find the maximum convex surface of a cylinder inscribed in a 
cone whose altitude is 6, and the radius of whose base is a. 

Ans. Maximum surface = 7cdb/2. 

15. Find the altitude of the cylinder of maximum convex surface that 
can be inscribed in a given sphere whose radius is r. 

Ans. Altitude = r V2. 

16. Find the altitude of the cone of maximum convex surface that can 
be inscribed in a given sphere whose radius is r. 

Ans. Altitude = 4 r/ 3. 



MAXIMA AND MINIMA. 



91 



17. A privateer has to pass between two lights, A and B, on opposite 
headlands. The intensity of each light is known, and also the distance 
between them. At what point nmst the privateer cross the line joining 
the lights so as to be in the light as little as i^ossible ? 

Let c = the distance AB, 
and X = the distance from A to any point P on AB. 

Let a and b be the intensities of the lights A and 1>, respectively, 
at a unit's distance. The intensity of a light at any point equals its 
intensity at a unit's distance divided by the square of the distance 
of the point from the light. 

Hence, the function whose minimum we seek is 
a/x^ + b/{c — x)2. 

Ans. x = cai/V(a^/^ + ^^/'^)- 

18. The flame of a lamp is directly over the centre of a circle whose 
radius is r ; what is the distance of the flame above the centre when the 
circumference is illuminated as much as possible ? 

Let A be the flame, P any point on the circum- 
ference, and X = AC. The intensity of illumina- 
tion at P varies directly as sin CPA, and inversely 
as the square of PA. Hence, the function whose 
maximum is required is ax / if- + xP-y'^-, where r 
is the radius of the circle, and a is the intensity of 
illumination at a unit's distance from the flame. 

Am. X = /-VlV' 




19. On the line joining tlie centres of two spheres, find the point from 
which the maximuiu oi" s[)licrical surface^ is visible. 

Let cv = r, GP - R, cC. = a, 
and cA = x, A being any point 
on 7riM. From A draw the tan- 
gents Ap and A P ; then the 
sum of the zones whose alti- c n m 
tudes are iini and .V.U, rt'spec- 
tively, is the fnnctit)n wliosi> niaxinuim is reiiuirod. 

By geomc>lrv this fiuu-tion is 







92 DIFFERENTIAL CALCULUS. 

20. Assuming that tlie work of driving a steamer through the water 
varies as the cube of her speed, show that her most economical rate per 
hour against a current running c miles per hour is 3c/2 miles per 
hour. 

Let V = the speed of the steamer in miles per hour. 

Then av^ = the work per hour, a being a constant ; 

and V — c = the actual distance advanced per hour. 

Hence, av^ / {v — c) = the work per mile of actual advance. 

21. The amount of fuel consumed by a certain ocean steamer varies as 
the cube of her speed. When her speed is 15 miles per hour she con- 
sumes 4^ tons of coal per hour at $4 per ton. The other expenses are 
|12 per hour. Find her most economical speed and the minimum cost 
of a voyage of 2080 miles. ^^s. 10.4 miles per hour ; $3600. 

22. Find the parabola of minimum area which shall circumscribe a 
given circle whose radius is r. Ans. if- = rx. 

23. One dark night the captain of a man-of-war saw a privateersman 
crossing his path at right angles and at a distance ahead of c miles. The 
privateersman was making a miles an hour, while the man-of-war could 
make only b miles in the same time. The captain's only hope was to 
cross the track of the privateersman at as short a distance as possible 
under his stern, and to disable him by one or two well-directed shots ; 
so the ship's lights were put out and her course altered so as to effect 
this. Show that the man-of-war crossed the privateersman' s track 
(c/6) V(ci^ ~ ^^) miles astern of the latter. 

24. The limited line AB lies without and is oblique to the indefinite 
line CD ; find the point P in CD so that the angle APB will b e a maxi - 
mum. Ans. If AB produced meets CD in C, PC = ^AC ■ BC. 



CHAPTEE YIII. 

POINTS OF INFLEXION. CURVATURE. EVOLUTES. 

108. A curve is concave upward or downward at any point 
(x, y) according as d^y/dx^ is positive or negative. 





When a curve, as ah, is concave upward, tan <^ or dy j dx 
evidently increases when x increases ; hence, by Cor. of § 80, 
d^y /dx^ is positive. 

When a curve, as cd, is concave downward, tan c^ or dy /dx 
evidently decreases when x increases; hence, d^y/dx'-^ is neg- 
ative. 

109. A point of inflexion is a point, as P, where the tan- 
gent crosses the curve at the point of contact. 

On opposite sides of a point of inflex- 
ion, as P, the curve is concave in oppo- 
site directions, and d'^y /dx^ has opposite 
signs ; hence, at a i)oint of inflexion 
dy I dx has eitlier a maximum or a mini- 
mum value (§ 101). 

Therefore, to examine a curve for 
points of inflexion, we examine its sIojh^ dy /dx for maxima 
and minima. 




TIio \o\\(\ or ]ni(li wlnv^i^ 
inlloxioii, /*. 



;ulc is aPb is stcopest at tlio point of 



94 DIFFERENTIAL CALCULUS. 

EXAMPLES. 

1. Examine y = a -{- c {x -{- b)^ for points of inflexion. 

Here d-y /dx- = 6c{x + b). 

The root of 6 c (x + 6) = 

is — 6, and Qc{x + b) evidently changes from — to + when x passes 
through — b ; hence, (— 6, a) is a point of inflexion, or a point of 
minimum slope. To the right of (— b, a) the curve is concave upward. 

2. Examine x^ — 3 bx^ + a-y = for points of inflexion. 

Ans. (6, 2 W /oC-) is a point of inflexion, or of maximum slope, to 
the right of which the curve is concave downward. 

3. Examine ?/ = x'^ — 3 x^ — 9 x + 9 for points of inflexion. 

Ans. (1, — 2) is a point of inflexion, to the right of which the curve 
is concave upward. 

4. Examine y = c sin (x/a) for points of inflexion. 

Ans. (0, 0), {± an, 0), {± 2a;r, 0), • • •. 

6. Examine the witch of Agnesi ?/ = 8 a^ / (x^ + 4 or) for points of 
inflexion. Ans. (± 2aV3/3, 3a/2). 

6. Examine the curve y = x^ / (p? + x^) for points of inflexion. 

Ans. (0, 0), (aV3, 3aV3/4), (— a Vs, — 3aV3/4). 

110. Polar curves. From the figure it is evident that 
wlien a polar curve, as ab, is concave toward the pole, 7; or OD 
increases as p increases; hence, dp j dp is positive. 





When a curve, as cd^ is convex toward the pole, i^ decreases 
as p increases ; hence, djp J dp is negative. 



POINTS OF INFLEXION. 



95 



That is, a polar curve is concave or convex toward the ptole 
according as dp/d/o is positive or negative. 

At a point of inflexion on a polar curve, dp J dp changes its 
quality, and therefore ^ is a maximum or a minimum ; and 
conversely. Hence, to examine a polar curve for points of 
inflexion, we examine p for maxima and minima. 



Ex. Examine the lituus p-d = a- for points of inflexion. 
Here 



p = =r= 

Vp^ + {dp/dey^ 

dp _ 2 «2 (4 a^ — p4) 
dp 



2a^p 



V4 a4 + p4 



§ 63, (9) 



(4a4 + p4)3/2 

Hence, p = a v2 renders p a maximum; therefore, (aV2, 1/2) is a 
point of inflexion. 

In the logaritlimic spiral p — a^, dp /dp is always positive ; hence, the 
curve, being concave toward the pole at all points, has no point of 
inflexion. 

111. The curvature of any curve, as APQ (§ 112, fig.), at 
any point, as P, is the s-rate at which the curve bends at P, 
or the 5-rate at which the tangent revolves, where s denotes 
the lenerth of the variable arc AP. 



112. TJie curvature of a curve at (x, y) /.s^ d<^/ds radians 
to a unit ofs. 

Let AP = s, and let cj^ denote 
(in radians) the variable angle 
XAIF as P moves ah)ng the 
curve APQ; then, (>vidiMitly, 
the curvature of APQ at P 
equals the .<?-rate of </>, or 
d(f) I ds. 




O M 



113. Curvature of a circle. Let A/'Q be the arc o( a 
circh> whose radius is r; thiMi th(^ augh^ M/>\. ov Ac/), will 
equal the angle subtiMided bv I'Q, ov An, at. its centre. 



96 DIFEERENTIAL CALCULUS. 

Hence, by § 37, we have 

A<3f> = As/r; r . d4> / ds = 1 / r. §11 

That is, the curvature of a circle is constant, a7id equals 
1/r radians to a unit of arc. 

For example, if r = 5, the circle bends uniformly at tlie rate of 1 / 5 
radian to a unit of arc. 

If r = 1/3, the curvature of the circle is 3 radians per unit of arc. 

114. Circle of curvature. The curvature of any curve 
except the circle varies from one point to another. A circle 
tangent to a curve and having the same curvature as the curve 
at the point of contact is called the circle of curvature at that 
point; its radius is called the radius of curvature ; audits 
centre, the centre of curvature. 

Let U denote the radius of the circle of curvature at any 
point of a curve ; then, since the curvature of the curve, or 
d^/ds, equals the curvature of the circle, we have 
d<^ I ds = 1/i?, or i^ = ds I d^. 

If at P (§ 112, fig.) the direction of the path of (x, y) became constant, 
(x, y) would trace the tangent at P ; if at P the change of direction of 
the path became miiform with respect to s, (x, y) would trace the circle 
of curvature at P. 

115. To find E, in terms of x and y. 

ds/dx = [1 + (dij/dxyj'\ § 33, Cor. 1 

<f> = tB.ii-^(di//dx)', §33 

d^ _ d^ij /dx^ 
' 'Jx^\ + (dij/dxf' . 
. ds ^ [l+{dy/dx)^r\ 

dcf> dhjldx" ^ ^ 

B will be positive or negative according as d'^y /dx^ is posi- 
tive or negative ; that is, according as the curve is concave 
upward or downward. 

If we take the reciprocals of the members of (1), we obtain 
the curvature. 



RADII OF CURVATURE. 97 

EXAMPLES. 
Find B and the curvature of each of the following curves : 
1. The parabola y'^ = ipx. 

dy/dx = 2p/y, dhj / dx'^ = — Ap^-fy^. 
Substituting these values in (1) of § 115, we obtain 

■2/2 + 4p2s3/2 yZ _2(X+p)3/'^ 



\ y^ 



4 p'^ pi 



We neglect the quality of E, since the quality of d^y /dx'^ indicates 
whether the curve is concave upward or downward. 

At the vertex (0, 0), R = 2p, and the curvature is {l/2p) radian 
to a unit of arc. 

2. The equilateral hyperbola 2 xy = a'^. E = {x^ + ?/2)3 /2/a2. 

3 The ellime - + ^ - 1 ^- —~ 

6. Ihe ellipse ^^ + ^ - 1. ^^ - ^^^^.^ ^ ^^^.^^3^, 

The maximum curvature is a/b'^, and the minimum b/a"^. 

4. The curve y = x^ — ix^-lS x^ at (0, 0). E = 1/36. 

5. Ihe logarithmic curve y = a^. -j- = — , , ,, ., , • 

° ds {ni- -\- y-y/- 

6. The cubical parabola ?/3 =; a'-^x. -^ = — — —-, .,, • 

^ ^ ds (9?/* + a^)3/2 



7. The cycloid x = r vers- ^ (///>') =]= v 2 r?/ — //-. i? = 2 V2 ry. 
At the highest point II = 4 r, or the maximum of R is 4 ?•. 

8. The catenary y = . (e-^'/" + e- •'"/"). 

9. The hypocycloid x-/'^ + //-/'' = ri-z'^ 

10. 'IMie curve x^ f- + //^ /- = a^ /-. 

11. 'I'hc curve ?/'^ = (i.r- + .r"'. 



(?0 _ 


(? 








C/N 


//■- 








/i' = 


•■' 


u- //)!/-. 






(/0 




d' /- 






(/.S 


2 (.1 


: + .v)«^- 






(Z0_ 




8.r-// 






ds 


In' 


+ (4.r + .r 


•->T 


fi 



98 



DIEFERENTIAL CALCULUS. 



116. To find E, in terms of p and 0. 
Erom (4) and (10) of § 63, we have 





'''Ye 



dp/dO 
dl d4 _ {dp/dOf- p.d^p/d6' 

do' de~ p'-^idp/doy 
J4> _ p^ + 2 {dp/dey - p ■ d'p ide^ 



• • dQ 



p' + {dp/dey 

. ds/de _ jp' + idp/deyj^' 
d<j>/de 



p'^-2{dp/dey- p-d^p/de' 



§ 63, (3) 



+ 2 

E = pVl + (loga)2. 



EXAMPLES. 
Find R in each of the following curves : 
i. The spiral of Archimedes p = ad. 

Here dp/dd = a, d^p /de^ = 0; 

„_ (o2 + a2)3/2 _a(^2+ 1)3/2 

p- + 2 a2 

2. The logarithinic spiral p = a^. 

3. The lemniscate p^ = a^ cos 2 6. B — €?■/?> p. 

4. The cardioid p = a(l — cos ^). U — 2V2ap/S. 

5. The curve p = a sec^ {d/2). R = 2a sec^ (^/2). 

117. Co-ordinates of centre of curvature. Let F(x, y) 

be any point on the curve ah, 
and (7 (a, ^) the corresponding 
centre of curvature. 

Then FC equals R and is per- 
pendicular to the tangent PD. 

Hence, ABCF = A XDF 

0A= OE - BF, 
AC-=EF-{-BC) 




EVOLUTES AND INVOLUTES. 



or 



X- R 



that is, a — X — R sin <^, ft = y -\- R cos <^ ; 

Substituting in (2) the values of R and ds^ we have 

Ay 1^ 

i8 = y + 



[ 



1 + 



dx 



\dxj 



d^y I dx^ 



d^y I dx? 



99 

(1) 

(2) 

(3) 




118. Evolutes and involutes. If the point (x, ij) moves 
along the curve J/iV, by equa- 
tions (3) of § 117 the point y 
(a, 13) will trace some other 
curve, as AB. The curve AB, 
which is the locus of the cen- 
tres of curvature of MN, is 
called the evolufe of MN. 

To express the inverse rela- 
tion, MN is called the involute 
of AB. ^ -^ 

119. Properties of the involute and evolute. 

I. From Cor. 1 of § 33 and ds = Rd(l>, we have 

dx = cos cfi ds = R cos <^ d(f>, (1) 

and dy = sin <^ (/s = R sin <^ ^/<^. (2) 

Differentiating ecjuations (1) of i^ 1 1 7, ami using tlu^ rela- 
tions given in (1) and (2), we obtain 

da = dx — R cos cf>d(f) — sin <f)dli = — sin (fydR, i^o) 
d(3 = dy — R sin <^ dcf> + cos cf) d/i = cos <^ dR. (4) 

Dividing (4) by (3), we obtain 

dft/da = — cot <f) = — dx / dy. 

Tliat is, fill' nornid/ to tJic inrolutc of {\. _v\ as P, (^^j 1 IS. tig.V 
ts tiDKjoit to the ccohitc at the correspond itnj point \^a, p\ as C\. 



100 



DIFFERENTIAL CALCULUS. 



II. Squaring and adding (3) and (4), we obtain 

da" + d^' = dR\ 
Let s denote the length of an arc of the evolute ; then 

da^ + dj^'' = ds\ 



ds 



dB 



.'. As = ±AE. 



Hence, 

That is, the difference between tivo radii of curvature, as 
C3P3 and CiPi (§ 118, fig.), is equal to the corresponding arc of 
the evolute, as C1C3. 

These two properties show that the involute 3IJV can be 
traced by a point in a string unwound from the evolute AB. 
From this property the evolute receives its name. 

120. To find the equation of the evolute of a given involute. 
Differentiating the equation of the involute and using equa- 
tions (3) of § 117, we obtain a and ^ in terms of x and y. 
These two equations and that of the involute form a system 
of three equations between a, ^, x, and y. 

Eliminating x and y from these equations, we obtain a rela- 
tion between a and /?, or the equation sought. 



EXAMPLES. 
Find the equation of the evolute of 

„ 1. The parabola ?/2 = 4px. 



(1) 




dy /dx = '2p/y, d-y/dx^ = — 4:p^/y^. 

Substituting these values in (3) of § 117, 
we obtain 

a = 3x-\-2p; ^ = — y^/4^p'^; 
.-. x = {a — 2p)/S, 2/ = — V4 j3p'^. 

Substituting these values of x and y in (1), 
we obtain 

/32-4(ar-2p)V27i9, (2) 

as the equation of the evolute of (1). 

The locus of (2) is the semi-cubical parab- 



EQUATION OF INVOLUTE. 



101 



ola. Thus, if nom is the locus of (1), F being the focus, then HAB is 
the locus of (2), where OA=2p = 2- OF. 



2. The ellipse a^y^ + fe^^ = aW. 

,2/3 / f)ip \2/3 



Here 



X- 






Hence, the equation of the evolute of (1) is 

(aa)2/3 + (6/3)2/3 = (^2 - 52)2/3. 



(1) 



§ 155, fig. 



3. The cycloid x = r vers-i {ij/r)zf V2 nj — y'^. 



dy _ V2 ry — y'^ d^ r_ _ 

dx y dx'^ ?/2 

•. ?/ = — /3, X = a — 2V— 2r/3 — i82. 

•. a = r vers-i (— /S/r) ± V— 2 r/3 — /32. 



(1) 




The locus of (1) is another cycloid tniual to the given cycloid, the 
highest point being at the origin. 

That is, the evolute of a cycloid is an equal cycloid. 

Thus, the evolute of the arc 07? is the arc OS, which equals RX ; 
and the evolute of RX is 8.Y, which equals OR. 

4. The hyperbola b-x- — a-y- = a-b-. 

5. Find the length of one branch of the cycloid. 
Here R = 2 V2r7/ ; .-. 8 A* = 4 r. 

ORX = 2 • OS = 2 N/i' = 8 r. 

6. Tlie length of (he evolute of the ellipse is \ {a'^ — b'^)/ab. 

Find iowv times the difference between /»' at (0, b) and li at {a. 0). 



102 DIFFERENTIAL CALCULUS. 

7. Find tlie length of an arc of the evolute of the parabola y'^ = 4:px 
in terms of the abscissas of its extremities. 

Arc AC =CP-AO= ^"^ ^ ^^ 2p Example 1, fig. 

_ 2 / g + j) 



[^-^y"-2v. 



8. Show that in the catenary ?/ = - (e^/« + e-^/«), 

a — x — - V?/2 _ ^2, p = 2y. 

9, Find the centre of curvature, and the equation of the evolute, of 
the hypocycloid x^^^ + y^/^ = a-^^. 

Ans. a = x + 3xi/3?/2/3; ^ = 2/ + 3x2/3?/i/3 ; 
(a 4-/3)2/3 4- (Q._^)2/3 = 2a2/3. 



CHAPTER IX. 



ENVELOPES. ORDEK OF CONTACT. OSCULATING CURVES. 



121. Family of curves. If in tlie equation, 

fix, y, a) = 0, 
different values are assigned to a, the resulting equations will 
represent a series of curves differing in position or form, but 
all belonging to the same class or family of curves. 

For example, if different values are assigned to a in tlie equation, 
(X - a)2 + if- = r2, 
the loci of the resulting equations will be a series of circles all having their 
centres on the x-axis and the same radius r (§ 122, fig.). 

As used in this chapter, the word curve includes the straight line. 

The quantity a, which is constant for the same curve, but 
different for different curves, is called the -paTameter of the 
family. Any two curves of the family which correspond to 
nearly equal values of a are called consecutive curves. 

Of the families considered in this chapter the consecutive 
curves intersect. 

122. Envelopes. If when consecutive curves approach 
indefinitely near each other, 
their points of intersection 
approach limits, the locus of 
these limits is called the envel- 
ope of the family. 

For example, if a is a variabli' 
parameter, the envcU>iH> of tlu' fam- 
ily of curves roi^rcsontiHl by llu' iHiuation, 

(.(• — .0- + //- -^ iT), 
is evideii(l> Mit> Iwo liiu>s ?/ = ± 5. 

This envelope is ovidoutly tangent to each curve of ihc family 




104 DIFFERENTIAL CALCULUS. 

123. The envelope is tangent to each curve of the family. 
Let A, B, C represent three consecutive curves of the family. 
Let P be the point of intersection of the curves A and B, and 
Q that of B and C. Conceive the curves A and C to approach 
the curve B, so that arc PQ = 0; then, since the limits of 



P and Q are on both the envelope and the curve B, the limit 
of the secant 3IPQN^\\\ be a common tangent to the envelope 
and the curve B. Hence, the envelope is tangent to the curve 
B at their common point. 

124. To find the equation of the envelope of a family of 

curves. 

Let fix, y, a) = 0, (1) 

and f(x, y, a + Aa) = 0, (2) 

be the equations of any two consecutive curves. 

Subtracting (1) from (2) and dividing by Aa, we have 

fix, y, a + Aa)-/(a;, y, a) ^ ^ 

Aa * ^ ^ 

By algebra the intersections of (1) and (3) are the same as 
those of (1) and (2). 

Making Aa = 0, from (3) we obtain (For notation see § 133) 

^J{x,y,a) = 0. (4) 

The intersections of (1) and (4) are, therefore, the limUs of 
the intersections of (1) and (2). Eliminating a between (1) 
and (4), we obtain the equation of the envelope. 



EQUATIONS OF ENVELOPES. 105 

EXAMPLES. 

1. Find the envelope of the family of straight lines represented by the 
equation y = ax + m/ a. (1) 

Differentiating (1) with respect to cc, we have 

= x-m/a\ (2) 

Eliminating a between (1) and (2), we obtain 

y2 = 4 mx ; (.3) 

that is, the envelope is the parabola (3). 

2. Find the envelope of the hypotenuse of the right-angled triangles 
which have the constant area c. 

Let a and j8 denote the lengths of the sides of the right triangles, 
and assume these sides as the co-ordhiate axes ; then we have 

x/a + y/p = \, ap = 2c. (1) 

Eliminating j8 between equations (1), we obtain 

x/a + ay /2 c = 1. (2) 

Differentiating (2) with respect to a, we have 

-x/a^-{-y/2c = 0. (3) 

Eliminating a between (2) and (3), we obtain 
xy = c/2; 
that is, the envelope is an hyperbola to which the sides of the tri- 
angle are asymptotes. 

3. Find the envelope of a line of constant length c whose extremities 
move along two fixed rectangular axes. 

Let a and ]8 be the intercepts of the line on the axes ; then 

x/a + ?///3 = l, a- + (3-^ = c\ (1) 

Differentiating eciuations (1), we obtain 

Dividing the first of equations (2) by the second, wo have 
x/a _ yjj^ _ x/a + y/^ _ \_ 
a- iS- a- + {i- C' 

Substituting these \aluos in imMut oi equations (1), we tind tho 
envelope to be the li\ luu-ycloid .f'-^^ + //'-^^ = c-'-'. (§ loo, tig. 8.) 



106 DIFFEREXTIAL CALCULUS. 

4. Find tlie envelope of the family of lines whose equation is 
a and /3 having the relation ex /I + ^/in = 1. 

Alls. (x//)l/2 + (y/„;)l/2- 1, 

5. Find the envelope of the family of ellipses defined by the equations 

x^/a^ + y^/^2 = 1, a^/m^ + ^^/n^ = 1. 

Ans. The four lines :L x/m ^y/n = 1. 

6. Find the envelope of the family of right lines, 

y = ax± -^aP-ofi — 6-, (1) 

where a is the variable parameter. 

Here 0=x±— ===; .-. or = ip — -==^ • (2) 

Substituting this value of a in (1), we obtain 

h (x- — a-) 6 /— ; : 

?/ = =F - / , = q= - -V x^ - a^ 

a Vx'- — Or a 

or x-/a2 — y- /h- — 1. (3) 

In equations (1) and (2) the upper signs go together. 
Here, as in example 1, the equation of the tangent is given and 
that of the curve is required ; hence, the method of envelopes has 
sometimes been called "the inverse method of tangents." 

7. Find the envelope of the family of parabolas ?/2 — ^^i^x — a)., a 
being a variable parameter. ^,^5_ 2/ = ± x/2. 

8. Find the envelope of the family of circles whose diameters are the 
double ordinates of the parabola ij- — 4px. . Ans. y~ = 4_p (p + x). 

9. Find the envelope of the family of circles whose diameters are the 
double ordinates of the ellipse x^ /a- + y'^/W' = 1. 

Ans. xV(a- + &-) + 2/V^- = 1- 

10, Show that the envelope of the normals to any curve, MN (§ 118, 
fig,), is the evolute AB of that curve. 

Let P2 approach P3 as its limit ; then the intersection of the nor- 
mals P2C2 and P^C^ will evidently approach C3 as its limit. 
Hence, the evolute AB is the envelope of the normals to MN. 



EQUATIONS OF ENVELOPES. 



107 



11. Using the principle in example 10, find the evolute of the parabola 
y^ = ipx, having given the equation of the normal in the form 

y = a{x — 2p) — pa^. 

12. Find the envelope of the family of ellipses 

x^/a^ + y^/{k — ay^ = 1, 
a being a variable parameter. Ayis. x^ ^^ -|- ^2 /3 — ^2 /s^ 

13. Find the envelope of the family of parabolas 

y = ax — {1 — a^)x^/2p, 
a being a variable parameter. Ans. x- = — p [p + 2y). 




125. Different orders of contact. Let APB and CFB be 

any two curves, and let F, Q, B be three of their points of 

intersection. Suppose 

that Q moves toward P 

and becomes coincident 

with it. The curves are 

then said to have contact 

of the first order at P. 

At such a point the 
curves do not cross each 
other, since CF and QH "-' 
are on the same side of 
AFB. 

Again, suppose that Q and B both become coincident with 
F. Three points of intersection will then coincide at P, and 
the curves are said to have contact of the second order. 

At such a point the curves cross each other, since CF and 
BD are on opposite sides of AFB. 

If four points of intersection become coincident at /\ tlie 
contact is of the third order, and tlie curves do not cross at J\ 

In general, if (k + 1) poi/its of interserfiofi coifiritfe at V, 
the contact is said to he oj' (he Vtli order, (uu/ the en r res will, or 
will not, ei'oss at 1* aeeordi/nj as the order of' eo/daet is orcn 
or odd. 



108 



DIFFERENTIAL CALCULUS. 



^=/^ 




126. Analytic conditions for contact of the kth order. 

Let tlie curves y =fx and y = <^x intersect in the points P^ 

Let OJfi = a, and let 
h denote the distances 
from ilfi to the ordinates 

that is, let 
^ ^^^^ ^^ ^^^^ h = 0, M^M^, M^M„ ' • •. 

Then, for each of these values of h, we have 

(f,(a-\- h) =f{a + h), or =f{a + A) - c^ (a + h). (1) 

Expanding the second member of (1) by Taylor.'s theorem, 
we have 

= {fa- ^a) -h {fa - c^'a) h + (/"a - c^"«) ~ 

+ (fa - r'») I . + ... + (/„ - 4,^,) ^ + • • -. (2) 

If P2 coincides with Pi, that is, if two values of h are zero ; 
by the theory of equations, from (2) we have 

fa = cf^a, fa = cf^'a. (3) 

Hence, (3) are the two conditions for contact of the first 
order. 

If three values of h are zero, from (2) we have 

fa = <^r/, fa = cj^'a, f'a = <j)"a. (4) 

Hence, (4) are the three conditions for contact of the second 
order. 

In general, if A; + 1 values of h are zero, from (2) we have 

fa = <^a, fa = cfi'a, f'a = ({y"a, • • •, fa = <^^a. (5) 

Hence, (5) are the k + 1 conditions for contact of the ^th 
order. 



OSCULATING CURVES. 109 

Or, in the differential notation, if when x = a 

y, dy /dx, d'^y /dx^, • • •, d^y / dx^^ 

all have the same values in one of two equations as in the 
other, the loci of these equations have contact of the 7cth 
order at (a, y). 

Ex. Find the values of a, 6, c when the curves 

y — (xx2 + 6x + c and ij — log {x — 3) 
have contact of the second order at the point (4, 0). 
From y = log {x — 3), we have when x = 4, 

y = 0, dy/dx = 1, d^y/dx'^ = — 1. (1) 

From y = ax'^ + 5x + c, we have when x = 4, 

2/ = lGa + 46 + c, dy/dx = ^a + h, dhj / dx''- = 2 a. (2) 

Equating the values of ?/, dy /dx^ and d-y /dxr in (1) and (2), we 
obtain 

16a + 46 + c = 0, 8a + 6=], 2a =-1. (3) 

Solving system (3), we obtain 

a=-l/2, 6 = 5, c=:-12. 

Hence, the curve y = — x2/2 + 5x — 12 has contact of the second 
order with the curve y = log(x — 3) at the point (4, 0). 

Only three independent conditions can be imposed on the t iiree general 
constants a, b, c ; hence, in general, the given curves cannot have an 
order of contact above the second. 

127. Osculating curves. The straight line of closest contact 
(a tangent) has, in general, contact of the first order ; for two 
and only two indepeiulent conditions can be nn}n>sed upon 
tlie two arbitrary constants in the goiuM-nl linear equation, 

// = inx + V. 

The elvch of closest contact, called the osculatinfj circle, has, 
in general, contact of the second order ; for three and only 
three independent conditions can be inipos^nl upon tlie three 
arbitrary constants in the genm-al ecpiatien of {\w eirch>. 

{X a)- -h {^y />)- : ;•-. 



110 DIFFERENTIAL CALCULUS. 

The parabola of closest contact, called the oscillating parabola, 
has, .in general, contact of the third order; for the general 
equation of the parabola has four arbitrary constants. 

The conic of closest contact, called the osculating conic, has, 
in general, contact of the fourth order ; for the general equa- 
tion of the conic has five arbitrary constants. 

It was necessary to quahfy the above propositions by the words ' in 
general ' ; for at particular points the contact may be of a higher order 
than at points in general. For example, at a point of maximum or mini- 
mum slope the tangent has three coincident points in common with the 
curve, the contact is of the second order, and the tangent crosses the 
curve (§ 109). Again, at a point of maximum or minimum curvature, 
the circle of curvature, which does not cross the curve at the point of 
contact (§ 130), has contact of the third order at least, 

128. The osculating circle is the circle of curvature. 

From §§ 115 and 126 it follows that any two curves which 
have contact of the second order have the same curvature at 
their common point ; and conversely. 

129. The circle of curvature, in general, crosses the curve at 
the point of contact. 

For its contact is, in general, of an even order. 

130. ^At a point of maximum or minimum curvature, the 
circle of curvature does not cross the curve. 

For on each side of a point of maximum curvature, the curve 
changes its direction more slowly than at this point ; hence, 
on each side of this point, the curve lies without the circle of 
curvature at this point, and therefore does not cross it. 

For a similar reason, the circle of curvature at a point of 
minimum curvature does not cross the curve. 

Since the circle of curvature at a point of maximum or 
minimum curvature does not cross the curve, the contact 
must be of an odd order (the third at least). 

* By a maximum or a minimum curvature is meant an arithmetic 
maximum or minimum, the quality of the curvature not being considered. 



ORDERS OF CONTACT. m 



EXAMPLES. 

1. By drawing the figures, show that the four intersections of a circle 
and an ellipse coincide when the circle becomes the osculating circle to 
the ellipse at either end of the major or the minor axis. 

2. Show that the curve 4 ?/ = 8 x^ — x^ and the line 4 ?/ = .3 x — 1 have 
contact of the second order. 

3. Show that the parabola 8 ?/ = x^ — 8 and the circle x^ + ?/2 = 6 ?/ + 7 
have contact of the third order. 

4. Find the order of contact of the hyperbola xy —\ and the i3arabola 

(x-2)2 + (?/-2)2 = 2x?/. 

5. Find the value of a when the hyperbola x?/ = 3x — 1 and the 
parabola y = x -\- \ -\- a{x — Vf- have contact of the second order. 

Ans. a = — 1. 

6. Find the values of m and c when y — mx + c has contact of the 
second order with y = x^ — 3 x^ — 9 x + 0. ^,js. m = — 12 c = 10. 

7. Find the values of a, 6, c when the curves 

y = x^ and y = ax- + bx + c 
have contact of the second order at the point (1, 1). 

8. Find the values of a, b, c when the curves 

?/ = sin X and y = ax- + bx + c 
have contact of the third order at {tt /'I, 1). 

Ans. « = — 1/2, b---7t/2, c = 1 — ;r-/8. 



CHAPTEE X. 
CHANGE OF THE INDEPENDENT VAEIABLE. 

131. Different forms of the successive derivatives of 
dy/dx. 

(i) When x is independent, by § 78 we liave 
d dy _ d^y d d dy _ d^y 
dx dx dx^ dx dx dx dx^ 

(ii) Wlien neither x nor y is independent, dy/dx is a 
fraction having a variable numerator and a variable denomi- 
nator, and d dx = d?x, etc. ; hence, 

d dy _ dxdhj — dyd'^x 

dx dx dx? ^ ^ 

d_ d_dy__ dx'dhj - dxdyd^x - 3 dxd^xdhi + 3 dyid'-xf 
dx dx dx dx^ ' ^''^ 

(iii) When y is independent, d^y = 0, d^y = 0, • • • ; hence, 
from (1) and (2) we obtain 



d dy _ dyd^x 
dx dx dx^ 



(1') 



d d dy __S dy (d^xy — dxdyd^x 



dx dx dx dx 



(2') 



132. Change of the independent variable. In the appli- 
cations of the Calculus it is sometimes necessary to make a 
differential equation depend on a new independent variable 
instead of the one which was originally selected ; that is, we 
need to cha7ige the independent variable. 



CHANGE OF THE INDEPENDENT VAEIABLE. 113 

When X = cfi (z) and we wisli to cliange the independent 
variable from x to z, we substitute for d'^y / dx^, dhj /d/jf', • • •, 
respectively, the second members of (1), (2), • • •, in § 131. 

In the resulting equation we substitute for x, dx, d^x, • • •, 
their values as obtained from the equation x = ^ (z). 

Ex. 1. Given x — cos 6, change the independent variable from x to 
e in 

^_^L_^ + J_:^0. (1) 

dx^ 1 — x'^ dx 1 — x'^ ^ ' 

Substituting for d'^y/dx'^ the second member of (1) in § 131, we have 

dxd'^y — dyd^x _ x dy y _ 

dx^ 1 - x2 dx 1 - x2 ^ ' 

X = cos 0, .-. dx = — sin 6d9^ 

d^x = — cos edd^, 1 — x2 = sin2 d. 

Substituting these values in (2) and simplifying, we obtain 

d-^y/dd'^ + ?/ = 0. (3) 

Equation (3) is the differential equation which would have been 
obtained if the differentiations which led to (1) when x was independent 
had been performed on the same equation under the new hypothesis 
that X = cos d and 6 is independent. 

Ex. 2. Given y = tan z, change the function from y to z in 

d'y^. I 2(1+7/) /dy\\ 

dx2 "^ 1+2/2 \ax) ^^ 

dy „ dz 

y = tan z, .-. -7- = sec^z -t" ' 

dx dx 

^ = 2 sec^^; tan z{ — ] -\- scc-z 3-T,' 
dx- V (/.)•/ dx^ 

1 + ?/ = 1 + tan Zj 1 + y- = sec^z. 
Substituting these values in (1), wo obtiiin 

d-z/dx:' — 2 {dz/dx)~ = cos'^z. 

To change the indcponilcnt variablo from .r to // wo substi- 
tute for d'v/ /dx\ d'\// /d.r\ • • •, rospin'tivoly. the last luombors 
of (1'), (2'), • • •, in § 131. 



114 . DIFFERENTIAL CALCULUS. 

Ex. 3. Change the independent variable from x to ?/ in 

Kdx^J dx dx^ dx^Kdx) ' ^' 

Substituting in (1) for d^y/dx^ and d^y/dx^, respectively, the last 
members of (!') and (2') in § 131, we obtain an equation which may be 
reduced to the form 

d^x/dy^ + d^x/dy^ = 0. (2) 

The position of dy in (2) indicates that y is independent. 

EXAMPLES. 
Change the independent variable from x to 2 in 

Atis. d^y/dz'^ + (a - 1) (dy/dz) + l)y = 0. 

2. x2f| + 2x^ + ^2/ = 0, given x = - • Ans. f| + a^y = 0. 

dx^ dx x^ ^ z dz^ 

3. (1 — x2) -— : = X -^^ given x = cos z. • Ans. -— ; = 0. 
^ ' dx^ dx ^ dz^ 

. d-y . 2x dy . y . , 

4- T^ + T—. — - -^ + ,., , .,,., = 0, given x = tan z. 
dx^ \ + x^ dx (1 + x^y ° 

Ans. d'^y/dz'^ + 2/ = 0. 
^ d^y . 1 dy . . . „ , . d^y . dy . _ 



Change the independent variable from x to ?/ in 

d2x 
dy^ 



\ dx J\dx^/ \ dx J dx dx^ 

. /d^x\^ /dx , \ dH 

8. Change the independent variable from x to ^ in 

„ [l + (cZ?//dx)213/2 . . • z, 

B = ^-^-^ — — , given X = p cos 6j y = p sin 0, 

d^y/dx"^ 

p being a function of 6. Ans. The value of E in § 116. 



CHAPTEE XI. 
FUNCTIONS OF TWO OR MORE VARIABLES. 

133. Partial differentials and derivatives. 

Let u=f{x,y), 

where x and y are both independent. The differential of u as 
a function of x, y being regarded as constant, is denoted by 
Q^u ; and the differential of u when y alone is variable is 
denoted by "dyU. These differentials are called the partial 
differentials of u with respect to x and ?/, respectively. 

The partial derivatives of u with respect to x and y are 
denoted by Qu / dx and 'du/dy, respectively. 

For example, if u — x'^/o?' + if- /^^^ 

dii /dx = 2x/ a2, 3u/di/ = 22// 6-. 

xdu . y du _ XT' if _ 
" 2 dx 2 dy ~ d^ W- 

EXAMPLES. 

1. When u = hifx + cx^ + gif + r.r, 

9,,it = (bif + 2 ex + e) dx, d,,}i = {'2 bxi/ + il (///-') (///. 

2. u = logx", X ■ dii/dx + du /di/ = y + loi;-.r, 

3. ((. = ?/■'•, 3;/ /(?x + (///.(•) • Bu /dij = u (loo- // + 1). 

4. w = log(c»' + t'."), 9«V(?x + du/di/ = \. 

5. i/, = log {x + Vx- + //-), X • 9u /(/.r + ?/ • du/dy = 1. 

6. ?A = xy/{x + ?/), .r • du /dx + // ■ du/dy = u. 

7. K. = x"y'\ X ■ du /dx + // • du/dy — (.r + // + log u) u. 



116 



DIFFERENTIAL CALCULUS. 



134. Total differentials. If we differentiate u = f{x, y), 
supposing X and y both to vary, we obtain the total differ- 
ential, dii, or df(x, y). 

The proofs in §§ 16-28 hold equally well when u, y, and z 
denote functions of two or more independent variables ; hence, 
the total differential of f(x, y) may be obtained by the prin- 
ciples in those articles. 

135. The total differential of a function of two or "inore 
variables is equal to the sum of its partial differentials. 

By §§ 16-28 we know that all the terms of df{x, y) are 
linear in dx and dy. Hence, if u =f{x, y), we may write 

du = 4> (x, y) dx + <^i {x, y) dy, (1) 

where <^ (x, y) and <^i (x, y) denote, respectively, the sums of 
the coefficients of dx and dy in the different terms of du. 
When x alone varies, (1) becomes 

Q,u = 4> (x, 7j) dx. (2) 

When y alone varies, (1) becomes 

ByU = <^i {x, y) dy, (3) 

From (1), (2), and (3), we obtain 

du = S^u -h Syit. 

To illustrate this theorem geometrically, let P(x, y) be a moving point 
Y in the first quadrant XOY, x and y both 

jf I (5 being independent. 

Let CD and AH, respectively, repre- 
sent what Ax and Ay would be, if at the 
values OC and OA the change of each x 
and y became uniform with respect to 
the same variable ; then CD = dx and 
AH= dy. 
Let u = area of rectangle OCPA = xy ; 

then QxU = CDFP = ydx, 

dyU = APGH = xdy, 
and du = CDFP + APGH 



dy 


C 


F 






P 




y 


X 




dx 





c 



D X 



IMPLICIT FUNCTIONS. 117 

Notation. It is often convenient to denote 3.jc by — — dx 

or — - ti • dx, and Q„u by — — du or —- u- dy. 
dx ' " ^ dy -^ dy -^ 

If u =f(x, y) = a, by § 135, we have 

— — dx + — — di/ = du = 0. (2) 

dx dy ^ ^ 

Solving (2) for dy/dx, we obtain (1). 

Note. This formula for the derivative of an implicit function in terms 
of its partial derivatives is often useful and should be fixed in mind. 

Ex. Given y'^ — 2 x^y -\- bx = a = u, to find dy/dx. 
Here 3u/dx = — 4xy -{- b, 

and du/dy = Si/^-2x'^; 

.-. dy/dx = (4 xy -b)/ (3 ?/2 - 2 x2) . by (1) 

EXAMPLES. 
By § 135 find du when 

1. iL = bxy'^ + cx2 + <7?/3. du = {by- + 2 ex) dx + (2 bxy + 3 gy-) dy. 

2. u = y^'. dn = y^ log ydx + xy^'-'^ dy. 

3. u = logx". du = x-'^ y dx +• log x dy. 

4. u = tan-i {y /x). du ~ {xdy — ydx) / {x- 4- y-). 

5. u = ^y^i"''. du = //-»"•!• log// cos.r dx + ?/=^i»-' — i sin x dy. 
By § 13G find dy/dx when 

6. x"' + y/^ — 3 axy = 0. dy/dx = (.r- — a//) / ((T.r — //-). 

7. x">/a>" + y>"/b"> = 1. dy/dx = — {x / y)'»-^{b/a)"\ 

8. x-v — ?/'' = 0, dy/dx = {y- — xy log//) /(.r- — xy !og.r). 

9. X log y — y log x = 0. - - = 

dx X 1/ \0ixx — X 



118 DIFEERENTIAL CALCULUS. 

137. If u = f(x, y, z), y = cf> (x'), and z = ^i(x), u is directly 
a fiiuction of x and indirectly a function of x through y and z. 
The differentiation of such functions is often simplified by 
using the formulas in the next article. 

138. If u = f(x, y, z), y = <^(x), and z = <^i(x), 

du _ Sic 3u dy 3ic dz 
dx dx dy dx dz dx ^ "^ 

where du/dx is the total derivative of w. as a function ofx. 

au = ^clx + ^chj + ^d.. §135 

dx dy dz 

Dividing by dx, we obtain (1). 

CoPv. 1. If ii =f(ij, z), y = cf> (x), and z = <fi, (x), 

du _ Qu dy 3u dz 

dx dy dx dz dx ^ "^ 

Cor. 2. If u = f(i/) and y = 4>(x), -— = — - -^ • 

EXAMPLES. 

1. ii — z'^-\- y^ + zy, z = sin x, y = e-^ ; find du/dx. 

Here du/dy = Sy'^ + z, du/dz = 2z + y, 

dz/dx = cosx, dy /dx = e^. 

Substituting these values in (2) of § 138, we have 
du/dx = (3 y- + z) e^ -h {2 z + y) cos x 

= (3 e2a: + sin x) e"" + (2 sin x + e-'^) cos x 
= 3 e^^ + e^ (sin x + cos x) + sin 2 x. 

2. u — tan-i (x?/), y - e^. du/dx = e^ (1 + x) / (1 + x^e^^). 

3. u = e«'*' {ij — z), y = a sin x, z = cos x. du/dx = {a^ + 1) e«-^ sin x. 



4. w = tan-i-. a;2 + y2 = y2. 



du 1 



(Zx Vr2 



5. u = sm-1 z = e^, y = x^. — = (x — 2) — cos — ■ 



PARTIAL DIFFERENTIALS. 119 

^ ./^-j — 5 I *^^ ( 1 + m^) X + m c 

Q. u = V x^ + 2/^, 2/ = "ia: + c. — = • 



7. u — sin— 1 (?/ — 2), ?/ = 3 X, 2 = 4 x^. du/dx — Z/ Vl — x-, 

8. w = xV — x4?//2 + x*, ?/ = logic. du/dx = x3[4 (logx)^ + 7/2]. 

139. Partial differentials and derivatives of higher orders. 

If we suppose only one of the independent variables to vary 
at the same time, by successive differentiations we obtain the 
successive 2JCt^'tial dijferentials 8^^w, 'd'^yit, Q^^'^^j 9^2/'^^; * ' '» 

^'^*' -df'^'-'' rf^'^^' rf/^2/^. •••• 

For example, if u = x^ + xy- + ?/2, 

a,xW = (2 X + 2/2) dx, a2^M = 2 cZx2, 93^.?* = ; 

QyU = (2 x?/ + 2 ?/) #, 32^1* = (2 X + 2) cZ?/2, B^t = 0. 

If we differentiate u with respect to x, then this result 
with respect to y, we obtain the second partial differential, 

For example, if u = x^ + x2?/2, 

Qxit = (3 x2 + 2 X2/2) (7x, 92,.,,?/ = 4 xydxdi/. 

Similarly, the tliinl partial differential 9'\,/:^', or :~—yd//(I.r\ 

denotes the result obtained by dilferentiating // once with 
respect to y, then this result twice successively with respect 
to X. 

The symbols for the partial (/criratircs are 

3'"« 3''if 3''i( 3'''if 3''n 

dx^ dxdi/ d/f dx^ di/dx'-^ 

In iinding the succossivo partial dit'tVnMit ials and tUM-iva- 
tives of u, oi\f\x, //), Ave treat dx ami </// as iHtnstants. siut'o x 
and 1/ are independent variables. 



120 DIFFERENTIAL CALCULUS. 



140. If u =f(x, y), -^ = ^. (1) 

Qhi Bhi ^>hc ^ 

etc. ; (2) 



dx^dy dxdydx dydx'' 

that is, if u ^5 differentiated successively m times with respect 
to X and n times with respect to j, the result is independent of 
the order of these differ eiitiatioyis. 

Suppose X alone to vary in u =f{x, y); then 

A^2i =f(x + Ax, y) -fix, ?/). (3) 

Supposing y alone to vary in (3), we obtain 

^2/(^x^0 =/(^ -V Ax,y^ Ay) -f{x, y ^ Ay) 

-fix + Ax, ij) +f{x, y). 
In like manner we find 
A^iAyii) =fix + Ax,y + Ay) -fix + Ax, y) 

-fi^^^ y + ^y)+f{^, y)' 

.\A^(A^tt) = A^(A,/i), 

Ay \ Ax y Ax \ Ay J ^ ^ 

Every term in A^i j Ax which contains Ax vanishes in the 

limit ; asrain, every term in — ^ It -^ which contains A?/ van- 
' ^ ' "^ Ay Ax -^ 

A A u 
ishes in the limit : hence, every term in — ^ — ^- which con- 

tains either Ax or Ay vanishes in the limit. 

Likewise every term in -^ —^'-~ which contains either Ax 
"^ Ax Ay 

or Ay vanishes in the limit. 

Hence, by (4) we have 



SUCCESSIVE DERIVATIVES. 121 



Differentiating (1) with respect to x, we obtain 



(6) 



dx\dxdyj dx\dydxj^ dxdydx dydx'^ 
Applying the principle in (1) to 3u / dx, we have 






Bht 



dy dx\dx J ~ dx dy\dx J^ dx^dy ~ dxdydx 

From (5) and (6), we obtain (2); and so on. 

CoR. If, when Ax = i and A^ = vi, we have 

^^xtjU = (J3 (x, y) Aa^A?/ + vi'', where ?^ > 2 ; 
then a%^^ = <^{x, y) dxdy. (7) 

EXAMPLES. 

Verify the identities (1) and (2) of § 140 in each of the four following 
functions : 

1. u = cos {x + y). 3. u = tan— 1 {y/x). 

2. u = xhj- -\- mj'^. 4. u — sin (bx^ + ay'^). 

^ ' dx- dxdy dx 

_ x^y^ _ 8% 9^u _ ^ 9it 

X + y'' dxr- dxdy dx 

7. It u = {X- + ?/' 1 /-, x- v^ + 2 xy - + ?/- - -r, = 0. 

dx- dxdy dy~ 

8. If It - (.c + v/")^ /-, .t- -7-77 + 2 xy — — - + //- -—i = -u 

dx- dxdy dy- 4 



9. lfit = 



1 3-u , 9-'( , 9'-'/ 



r-)l/2' (f.j.J (/w2 (/~-J 



(.«;'-2 + //-^ + z-)i/'-^' (f.r- (/// 



10. If H = e'-."S ■ ^/", = (1 + '».n/.: + .r- //---) /^ 

dxdydz ^ 

11. li u = sm-^ixt/z), -, v~r = t 7~rV^^-r, " 



122 DIFFERENTIAL CALCULUS. 

141. To find the successive total differentials of a function of 
two independent variables in terms of its p)artial differentials. 

Let u = f(x, y) ; then, by § 135, we liave 

du = ——- dx H — — - chi. (V) 

dx dij ^ ^ 

.'. dhi = ——: dx^ -\r -, — T' dxdij + -— -— diidx + ——: dir, 
dx^ dxdy dydx dy 

ff,, ^^ax^ + 2^^ dxdy + || d,f. (2) 

Differentiating (2), remembering that, in general, each term 
in the second member is a function of both x and y, and apply- 
ing the principle of § 140, we obtain 

,P, = ^ a^ + 3 ^ ^^.^y + 3 _gL a,af + ^ d,/. 

By successive differentiations, we obtain d'^u, dhc, etc. 
Prom the analogy between these results and the binomial 
theorem, the formula for d^u is easily written out. 

142. Expansion of f (x + h, y 4- k). Eegarding x as the 
only variable, by Taylor's theorem we obtain 

fix + li,y + K) = f(x, y + k) + h~f(x, y + k) 
Regarding y as the only variable, we obtain 

7, pi 7^2 P)2 

f(x, y + k) =f(x, 2/) + J ^/(^. y) + |2 ^2/(-^^ !/)+•'•• 
Substituting in (1) this value off(x, y + ^), we obtain 
fix + h,y + k) =f(x, y) + ^^ ^/(^. V) + ^ ^/("^^ 2/) 



+|[''^£/(-'2')+2'^^^/(-'^)+^^U/(-'^)]+-' 



(C) 



TAYLOR'S THEOREM. 123 



A symbolic expression for this formula is 



/(^ + /., 2/ + k) =f(x, y) + (^i^^k dy)f{^. y) 



+|(^l + ^^^)/(^'^) + -"' (^') 



where \^^~^ — ^ ^ 



|3 V " dx ' " dy 

dx dy ^ 

is to be expanded by the binomial theorem and/(cc, y) written 
after each of the resulting terms. 

If u =f(x, y), we may put n for f(x, y) in (C) and (C). 

Compare (C) with (1) in § 144. 

CoR. 1. By a similar course of reasoning Taylor's theorem 
is extended to the expansion of functions of three or more 
independent variables, in series analogous to that in (C), 
or (C). 

CoR. 2. By Taylor's theorem we obtain the value of 
f(x + h)—f(x) in ascending powers of h; that is, Taylor's 
theorem expresses the increment of f(x) in ascending powers 
of the increment of x. Similarly, by the extension of Taylor's 
theorem, we express the increment of a function of two or 
more independent variables in ascending powers of the incre- 
ments of those variables. 

143. Maxima and minima of f(x, y). 

f(a, b) is a maximum of /(.r, //) whcMi, for all sninll pc^sitive 
or negative values of Jt and l\ 

f(a + //, />-\- k)-f((U i>)<0. 
f(a, f>) is ;i )t/hn')i/t()n of /'(.r, //) whcu, for all small positive 
or negative values of // and /,-, 

/{a + h. b -t- />•) -./V^ b')> 0. 



124 DIFFERENTIAL CALCULUS. 

144. Conditions for maxima and minima of f(x, y). 
Putting II =f{x, y), from (C) of § 142, we obtain 

12 I dx^ ^ dxdy "^ dy'^ J ' \ J 

When h and k are very small, the quality of 
f(x + h,y^ k) -fix, y), 
or of tlie second member of (1), will evidently depend upon 
the quality of h and k unless 

"du/dx = 0, and Qu / dy = 0. (a) 

But, by definition, when f(x, y) reaches a maximum or a 
minimum value, the quality of f(x -\- h, y -\- k) — f(x, y) is 
independent of h and k. Hence, equations (a) express one 
condition for a maximum or a minimum of f(x, y). 

Suppose x = a, y = b to \)Q one solution of system (a). 

Let A, B, C denote the values of 

Q^u/dx"^, B^f^/dxdy, 3^u/dy^, 
respectively, when x = a and y = h ; then from (1), we have 

f(a + A, ^ + k) -f(a, h) = i (Ah^ + 2 Bhk + Ck') -\- • - • 

_ (Ah -\- Bky ^ (AC - B') k' ^ ,„, 

- A\2 ^"'- ^^^ 

The quality of the second member of (2) is independent of 
h and k when and only when AC — B^ is positive or zero.* 
For when ^ C — ^^ is negative, the numerator will be positive 
when k = 0, and negative when Ah + Bk = 0. Hence, a 
second condition for a maximum or a minimum of f(x, y) is 

* The limits of this treatise exclude the investigation of the exceptional 
case when AC = B"^ or when ^ = 5 = C = 0. 



MAXIMA AND MINIMA. 125 

When condition (b) is satisfied by x = a, y = h\ from (2) 
we see that f(a, b) will be a maximum or a minimum of 
f(x, y) according as A^ or 'd'^u / dx^^^^, is negative or positive. 

CoR. 1. Condition (b) requires that Q'^u / dx^\j^j^ and 
B'^u / dif\^jj have the same quality. 

Note. This discussion assumes that /(cc, y) and all its successive 
derivatives are continuous functions. 

CoR. 2. By a similar course of reasoning we may obtain 
the conditions, for maxima and minima of functions of three 
or more variables. 

EXAMPLES. 
Examine for maxima and minima 

1. M =/(x, y) = x'^y + x?/2 — axy. 

3u/dx = {2x + y — a)y, du/di/ = (2y -{- x — a)x, 
d^u/dx^ — 2y, d-ii/dy- = 2x, 

Q'hi/dxdy = 2x4-2?/ — a. 
Hence, condition (a) of § 144 is 

(2 x + y- a) y = 0, (2 y + x- a) x = 0; (1) 

and condition (b) is 

4 x?/ >(2 X 4- 2 ?/ — a)2. (2) 

System (1) lias the four sohitions (0, 0), (a, 0), (0, a), (a/ 3, a/S). 

Tlie last, and it only, satisfies (2) ; lience, f{a/o, a /o) is a maxi- 
mum or a mininmm of /(x, ?/). 

If a is positive, d-u/dx'^ is iiositive when // = a/o ; lience, 
/(a/:], a/:]), or - aV^T, 
is a minimum. If a is ne,<;aliv(\ d'^u/dx- is negative when y = a/o; 
hence, —(V^/21 is a niaxinuun. 

2. V = x^ + //■' — ;> ax//. 

Aufi. When a is +, — «■> is a min. ; when a is — , — (V' is a max. 

3. u = x-^ + X!/ + //- — ax — by. j „,s.. ^ah - a- - l)-)/o is a min. 

4. u = x-hr{a -X- !/). ^.l,,,,.. (iV4;V2 is a max. 



126 DIFFERENTIAL CALCULUS. 

5. u = a:* + 2/*-2x2 + 4x?/-2?/2. ^^^^ - 8 is a min. 

6. w = (2ax-x2)(2 67/-2/2). ^^^_ a^b^ is Si msix. 

7. Find the maximum of X2/z subject to the condition 

Since xyz is arithmetically a maximum when x^y- • z^/c^ is a maxi- 
mum, we put 

Here condition (a) of § 144 is 
and condition (b) is 



The only solution of system (1) which satisfies (2) is 
X = a/ Vs, y = 6/ Vs. 

a^w ^ „/^ 6x2 7/2x 862 tt 5 

-7^ = 2 ?/2 ( 1 77, ) = rT i when x = -—= , y = --= 



Hence, a&c/3 Vs is a maximum of 



xyz. 



8. Given the sum of the three edges of a rectangular parallelopiped ; 
find its form when its volume is a maximum. Ans. A cube. 

9. Divide m into three parts x, y, z such that x"y^z^ may be a maxi- 
i^^iii- ^ns. x/a = y/b = z/c = m/{a + b + c). 

10. Divide 24 into three such parts that the continued product of the 
first, the square of the second, and the cube of the third may be a maxi- 
i^^m. Ans. 4, 8, 12. 

11. The vertices of a triangle are (Xi, ?/i), (X2, 2/2), and (X3, 2/3) ; find 
the point the sum of the squares of wiiose distances from the vertices is a 
minimum. 

Ans. [(xi + X2 + X3) /3, (?/i + 2/2 + 2/3) / 3], or the centre of gravity 
of the triangle. 



CHAPTER XII. 
ASYMPTOTES. SINGULAR POINTS. CURVE TRACING. 

145. An asymptote to a curve is a fixed line which is the 
limit of a tangent when the point of contact moves out along 
an infinite branch of the curve. 

146. To obtain the equations of the asym2)totes to the curve 

/(x, y) = 0, (1) 

where f (x, y) is of the nth degree in x and y. 

Let y = mx + I (2) 

be the equation of a tangent to (1). 

Substituting nix + I for y in (1), we obtain 

fix, mx + Z) = 0. (3) 

Equations (2) and (3) form a system which is equivalent 
to the system (1) and (2); hence, the n roots of (3) are the 
abscissas of the n points common to the curve (1) and its 
tangent (2). 

Since (2) is a tangent to (1), two roots of (3) are equal. 

Conceive the point of contact to move out along an infinite 
branch; then the two equal roots of (.'>) will beconu^ infinites. 

Therefore, by algebra,* the coefticients of .r" and .r"~' in (3) 
will approach zero as their common limit. 

* Substitutiiiii; 1 /.r for x in (1), and nniUiplyini;- by .r", wo obtain (^-J). 

X" + pi.r"- • + jj.a-"- - 4- • • • + i>„--j.r- + Pn-xx + i\, — 0. (H 

Vn'X''' + J)„_ i;c"-i + j);,_o.r"-- + • • • + jw-- -|- Pi.r + 1 ^ - 0. (2) 

The 71 roots of (2) arc the recipn>c'als of (ho n roots of (0, 

Whonp„ ■=! and j)„_i == 0, two roots of (1) =b 0; .-. two nnUs of (^"_M = X'. 

AVhon p„ = and Vn-\ — ^^ t'^^*^ roots of (0 = 0: .-. two of tho ); roots 
of ('2) nssnme tho form (i^;, and (2) has, in r(';;lity, only n — 2 roots. 



128 



DIFFERENTIAL CALCULUS. 



Hence, by putting the coefficients of a?" and cc"~^ equal to 
zero, and solving the resulting system of equations for m and 
I, we obtain the slope and the intercept of each asymptote. 

Ex. Examine for asymptotes the curve, 

y^ — ax^ — x^. (!') 

Let y = nix + I (2') 

be the equation of an asymptote to (1'). 

Substituting mx + I for y in (!') and arranging the terms according to 
the powers of x, we have 

(m3 + 1) a:3 + (8 mH - a) x2 + 3 ml^x + 1^ = 0. 




Putting the coefficients of x^ and x^ equal to zero, we have 

m3 + 1 = 0, SmH-a = 0. 
The only real solution of system (3') is m = — 1, 1= a/S. 
Substituting these values in (2^), we obtain 
y = —x + a/3, 
which is the only real asymptote to curve (!'). 

The locus of (!') is the curve nOPm, and that of (4') is the asymptote 
AB. 



(30 



(40 



CoR. 1. If we expand f(x, mx + l) and arrange the result 
according to the descending powers of x, (3) will assume the 
form 

A,x- -f A-i^""' + A-2^"~' + • • • + Aa^ + A = 0. (4) 

A^ will be of the nth degree in m, but will not contain I. 

■^n-i "^ill be linear in I. 



EQUATIONS OF ASYMPTOTES. 129 

Hence, tlie system, A^= and ^„_i = 0, (when determi- 
nate) has at most only n solutions ; whence a curve of the nth 
order cannot, i7i general, have more than n asymjjtotes. 

When, as is assumed in this article, equation (1) is of the 
nth degree in y, there will be no asymptote parallel to the 
2/-axis ; and conversely (§ 148). 

Con. 2. The n roots of equation (4) are the abscissas of 
the n points common to the curve (1) and the line (2). 

When A^= and A^_^ = 0, (4) is, in reality, of the 
(n — 2)th degree and has only n — 2 roots, the other two 
of the n roots assume the form ap. 

Hence, a7i asyuiptote to the cnrve (1) cannot have viore than 
n — 2 points in com,mon with the curve. 

Eor example, (1') and (4') form a system which is defective in two 
solutions; hence, the asymptote AB^ or (4'), has only the one point P 
in common -with the curve nOPm, or (1'). 

Any line parallel to an asymptote has a slope 7)i which 
satisfies A^ = ; but when J„ = 0, (4) has only 9i — 1 roots. 

Hence, any line parallel to an asymptote to the curve (1) can- 
not have more than n — 1 points in common with the curve. 

For example, (T) and ?/ = — x + c form a system which is defective in 
one solution ; hence, any line parallel to AB cannot have more than iwo 
points in compion with the curve nOPin. 

EXAMPLES. 
Examine for asymptotes 

1. 'llio folium of Descartes j'' + i/'^ = oad'y. § 155, fiij. (> 

2. ir ~ (LV- + x-\ y = x + a/'A. 

3. The conic sectitins. ay = ± bx. 

4. (//- — 1 ) // " i^x-2 — .J) .1-. y - X. 

5. //' — .r' + - iix~// -' />-.r-. 1/ = ±. X — a/'2. 



130 DIFFERENTIAL CALCULUS. 

6. x^ — y^ — a^xy — hhj^. y = ± x. 

7. y^ — Qxy~ + 11 x-y — Gx^ + X -\- y = 0. y = x, y = 2x,y = Sx. 

8. x^ -\- Sx-hj — xy^ - 3y^ + x'2 - 2xy -^ Sy^ + 4:x -\- b = 0. 

2/ = -a:/3-3/4, y = x + l/l, y=-x + S/2. 

9. Prove that the asymptote y = — x lacks three points of intersec- 
tion with its curve x^ + y^ = a^. 

147. Parallel asymptotes. In (4) of § 146 it sometimes 
happens that ^„_i = does not determine I for one or more 
of the values of m given by A^ = 0. If, in this case, I is 
determined by A^_2 = 0, wliich is a quadratic equation in I, 
we shall obtain for each value of m two values for I. This 
gives two ^^araZZeZ asymptotes, each of which lacks three 
points of intersection with its curve; for, in this case, (4) 
in § 146 has only 7i — 3 roots, the other three roots assume 
the form c^. 

If I is not determined by -4„_2 = 0, but is determined by 
A^_. = 0, we obtain three 2^(^^^allel asymptotes ; and so on. 

CoR. Any one of k parallel asymptotes lacks k + 1 points 
of intersection with its curve ; and any line parallel to k par- 
allel asymptotes lacks k points of intersection with its curve. 

Ex. Examine y^ — xy- — x-y ■\- x^ ■\- x- — y"^ — 1 for asymptotes. 
Substituting mx + I for ?/, we obtain 
(?n3 — m- — in + 1) x^ + (3 m-l — m- — 2 ml — Z + 1) x- 

+ (3 mW- — 2 ml - Z^) x + (l^ - Z^ _ i) = q. 

Equating the coefficients of x^ and x^ to zero, we liave 
m^ — mfi — ??i + 1 = 0, (1) T 

3 mH — m-2 — 2 7nl -1 + 1=0. (2) / 

From (1), ??i = — 1 or 1. 

When m = — 1, from (2) we obtain 1 = 0; hence, one asymptote is 
y=—x. 



ASYMPTOTES PARALLEL TO AN AXLS. 131 

When m = 1, (2) does not determine l. This indicates that there are 
parallel asymptotes having the slope 1. To obtain tlie values of I for 
these asymptotes, we equate the coefficient of x to zero, and obtain 

3 mZ2 -2ml-l^ = 0. (3) 

When m = 1, from (3) we obtain 1 = or 1 ; hence, the parallel 
asymptotes are y = x and y = x + 1. 

148. Asymptotes parallel to either axis. In § 146, let 

the axes be revolved until the ?/-axis is parallel to an asymp- 
tote ; then one value of m will assume the form op ; hence, 
A^^ = will be below the ?zth degree in m, and therefore 
/(ic, ^) — will be below the ?zth degree in y. 

Conversely, if /(cc, ^z) = is below the nth degree in ?/, there 
will be one or more asymptotes parallel to the ^/-axis. 

Hence, if /(cc, y) = is below the nt\\ degree in either x or 
y, there will be one or more asymptotes parallel to a co-ordi- 
nate axis. To find the equations of these asymptotes, equate 
to zero the coefficients of the highest jpoivers of x and y. 

The following example will make clear this principle : 

Ex. Find the asymptotes of the curve, 

?/2x2 — 3 ^2^2 _ 5 a.y2 _^ 2 x2 + 6 ?/2 + X + // + 1 = 0. (1 ) 

Arranging (1) in descending powers of a;, we have 

(?/- - 37/ + 2)a:2 - (5?/2 - i)a; + 6?/ + ,/ + i = q. ^o) 

E(iuating the coefficient of .x-' to zero, we obtain 

2/2 — 3 y + 2 = ; that^ is, // = 1 , // = 2. 

Substituting 1 for ?/, (2) becomes — .r + 2 = 0. 

Hence, (2) and ?/ = 1 form a system wliicli is defective in three sohi- 
tions ; that is, (2) and // =: 1 have only one conunon piMut. 

Substituting 2 for ?/, (2) becomes — 10x + 27=0; hence, (2) and 
?/ = 2 form a system which is defective hi three solutions. 

Therefore, ?/ = 1 and ?/ = 2 are two parallel asymptotes, which are 
jiarallel to the x-axis. 

Arranging (1) in cUvsccnding powers of //, we have 

(.r2 - 5 X + ('.) //2 - (:5 X- - U // + 2 .(-2 + .r + I = 0. (.">> 

I'^roiu (.".) \\t> st'i' that, .r — 2 and .r -- .". are two parallel as\ mptiUes. 
which art> parallel to the //-axis. 



132 



DIFFERENTIAL CALCULUS. 



EXAMPLES. 
Find the asymptotes to 

1. The cissoid of Diodes {2a — x)y^ = x^. 
See example 3 and fig. 3, § 155. 

2. The strophoid x (x2 + y^) + a {x'^ — y^) = 0. 
See example 7 and fig. 7, § 155. 

3. xy^ + x^y = a*. x = 0, y = 0. 

4. y■^x'^ - a^) = X. 6. x'-y^- = c^x'^ + y^). 

5. xy — cy — hx = 0. 7. {x — b)- {y — c) = a^. 

8. (x — a) ?/- = x-5 + aic2. a: = a, y = ±x±a. 

9. a:3 + 2 x2y + x?/2 — x2 — x?/ = — 2. x = 0, ?/ = — x, ?/ = — x + 1. 

10. y^ — x?/2 — x2?/ + x^ + x2 — ?/2 = 1. 2/ = ± a;, ?/ = X + 1. 

11. x2?/2 — x'^y — X?/2 + 2 X + 3 ?/ + 1 = 0. 

y = 0, y = I, X = 0, X = 1. 

12. ?/3 — 5 x^2 4- 8 x2?/ — 4 x3 — 3 ?/2 + 9 x?/ — 6 x2 + 2 2/ — 2 X = 1. 

?/ = x, ?/ = 2x + l, y = 2x + 2. 



149. Asymptotes to polar curves. When 
OF, or p, revolves about 0, it will become c^ 
when, and only when, it is parallel to an 
asymptote, as SK. Hence, any value of 
which makes p = cp gives the direction of 
an asymptote, and the corresponding value 
— of the subtangent OS gives the distance and 
direction of this asymptote from the pole 0. 

EXAMPLES. 
Examine for asymptotes 
1, The hyperbolic spiral p6 = a. 
Wlien ^ = 0, p = ap, and the subt. = — a. 

Therefore, the line parallel to the polar axis and at the distance a 
above it is an asymptote to the spiral. 




EQUATIONS OF ASYMPTOTES. 133 

2. The curve p cosd = a cos 2 6. 

When 6 = 7t/2, p = ap, and subt. = — a. 

Hence, the Ime perpendicular to the polar axis at the distance a to 
the left of the pole is an asymptote. 

3. The curve p^ cos 6 = a'^ sin 3 9. 

4. The curve p = a sec 2 6. 

There are four asymptotes forming the sides of a square, each 
being at the distance a/2 from the pole. 

5. The polar axis is an asymptote to the lituus pVd = a. 



Singular Points. 

150. The singular points of a curve are those points which 
have some peculiar property independent of the position of the 
co-ordinate axes. Points of inflexio7i and iriultiple points are 
varieties of singular points. 

Points of inflexion have already been considered in § 109. 

151. A multiple point is one through which two or more 
branches of a curve pass, or at wliich they meet. 




A multiple point is double when there are only two 
branches, triple when only thr(H\ and st) on. 

A multiple point of intersection is a unilliph^ point at 
which the branches cross (\ich other (tig. (A. 

An osculating point is a multi]de point through wliich tlie 
branches pass and at wliicli thi\v are tangiMit to each other 
(figs, h and c). 



134 DIFFERENTIAL CALCULUS. 

A cusp is a multiple point at which the branches terminate 
and are tangent to each other (figs, d and e). 

A cusp or osculating point is of the first or the second 
species according as the two branches are on opposite sides 
(figs, h and d) or the same side (figs, c and e) of their common 
tangent. 

152. To find the miolti^yle poi7its of a curve. 

At a multiple point each branch has its own tangent ; hence, 
at such a point dy / dx has two or more values. 

Let f{x, y) = be the algebraic equation of the curve free 
from radicals and fractions ; then, by § 136, we have 

-^ — — ^ , , f where u = f(x, ?/). 
dx Qu/dy J \ ■ J) 

Since u contains neither radicals nor fractions, this expres- 
sion for dy I dx can have only one value at any given point 
unless it assumes the indeterminate form 0/0; hence, at a 
multiple point we have 

Qu/dx = 0, 3u/dy = 0. (1) 

Any solution of system (1) which satisfies the equation of 
the curve gives a point on the curve at which 

dy/dx = 0/0. (2) 

The form in (2) can be evaluated by the method of § 88. 

If, in (2), dy I dx has two or more unequal real values, the 
point is, in general, a 7nultipl6 point of i7itersectio7i. 

If, in (2), dy /dx has two equal real values, the point is, in 
general, an osculating point or a cusp. 

Ex. 1. Examine the curve x* + ax'h/ — ay^ = for multiple points. 
Here u = x^ + ax^y — ay^ = ; (1) 

.'. Qu/dx = 4x^ + 2 axy, du/dy = ax^ — 3 ay^. (2) 

Equating these partial derivatives to zero, we have 

x{2x^ + ay) = 0, x^-3y^ = 0. (3) 



MULTIPLE POINTS. 135 

The only solution of system (3) whicli will satisfy (1) is x = 0, 2/ = ; 
hence, the only point to be examined is (0, 0). 
Prom (2) and § 136, we have 

cZy _ 4 x^ + 2 axy _ / x = 0, 

dx 3 ay'^ — ax^ 1^=0. 

Evaluating this fraction by the method of § 88, we find 
d7j/dx'\Q,o = 0, +1, — 1. 

Hence, the origin is a triple point of intersection at which the inclina- 
tions of the branches are, respectively, 0, 7r/4, 3 7r/4. 

The general form of the curve at the origin is shown in § 151, fig. a. 

Ex. 2. Examine the curve a*?/^ = a-x^ — x^ for multiple j)oints. 

Here u = a^if — a%4 + ^^e = Q; (1) 

.-. Sw/tZx = — 4 «2x3 4- G x^, Qu/dy = 2 a^- (2) 

The only solution of the system, 

- 4 a2x3 + 6 x5 = 0, 2 a4y = 0, (3) 

which will satisfy (1) is x = 0, ?/ = 0. 

dy 4 cC^x^ — 6 x5 , f x = 0, 
tx = —Ya^-' = r'^'''''\y = 0. 

Evaluating this fraction, we have 

d?//dx]o,o= ±0. 

An inspection of its equation shows that the curve passes through the 
origin and is symmetrical with respect to the x-axis ; hence, the origin is 
an osculating point of the first species (§ 155, fig. 5). 

Ex. 3. Examine y- = x(x + a)- for nuiltiple points. 

_, dy 8 X- + 4 ax + a- , f .r = — a, 

Here -7- = ^ = - when ■{ 

dx 2 // \y = 0. 

Evaluating this fraction, wo obtain 

(////(f.c]_,„o = ±^ — a. 

When a is negative, (— a, 0) is a double point of intersection. 

When a is ])ositive, tlie slopes of both tlio branches which {xiss through 
the point (— a, 0) are imaginary ; licncc, tlu>se brani'hcs do not lie in the 
plane of the axes, bnt are a i^art oi tlu> imaginary locus. 

When a is positive, tlu> multipK> in>int (— tf, 0) is isolatoil from the rest 
of the plane k)cus, and is called a co)ijiitju(c point. 



136 DIFFEKENTIAL CALCULUS. 

153. A conjugate point is a multiple point wliicli is formed 
by imaginary brandies meeting or crossing each other in the 
plane of the axes. 

Such a point is, in general, entirely isolated from the plane 
locus; but in exceptional cases it may lie on it. 

At a conjugate point dy /dx is, in general, imaginary; but in excep- 
tional cases it may be real, since the tangents to the imaginary branches 
at such a point may lie in the plane of the axes. 

A shooting point is a multiple point at which two or more branches 
end but are not tangent to each other. A stop point is a point at which 
a single branch of a curve ends. As a shooting point or a stop point 
never occurs on an algebraic curve, they will not be further considered, 

EXAMPLES. 
Show that the curve 

1. a?/2 = x3 has a cusp of the first species at (0, 0). § 155, fig. 2. 

2. ?/3 = 2 ax2 — x^ has a cusp of the first species at (0, 0). § 155, fig. 4. 

3. x^ + ?/^ = 3 axy has a double point of intersection at (0, 0). § 155, 
fig. 6. 

4. y^ = x^ {a^ — x2) has a double point of intersection at (0, 0). 

The form of this curve is similar to that of the curve in § 155, fig. 13. 

5. x{x~ + y-) = a {y- — x^) has a double point of intersection at (0, 0). 
§ 155, fig. 7. 

6. y'^ (a^ — x^) = X* has a point of osculation of the first species at 
(0, 0). 

7. 2/2 = 2 x^y + x'^y — 2 x* has a conjugate point at (0, 0). 

The slope of each branch at (0, 0) is real ; but (0, 0) is an isolated 
point ; hence, it must be a conjugate point at which the tangents to 
the imaginary branches lie in the plane of the axes. 

8. ay'^ = {x — a)2 (x — b) has at (a, 0) a conjugate point when a<Cb, 
a double point of intersection when a >> 6, and a cusp when a = b. 

9. a;^y^ — 2 abx^y = x^ has at (0, 0) a point of osculation and a point 
of inflexion on one branch. 



CURVE TRACING. 



137 



Curve Tracing. 

154. Symmetry. The following principles of symmetry 
of loci are easily proved : 

A locus is symmetrical with respect to the cc-axis when its 
equation contains only even powers of y. 

A locus is symmetrical with respect to the ^/-axis when its 
equation contains only even powers of x. 

A locus is symmetrical with respect to the origin when the 
terms in its equation are all of an odd or all of an even degree 
in X and y. 

155. To trace the locus of an equation we first find its 

Axis or centre of symmetry, if any ; 
Intercepts on the axes, and its limits ; 
Maxima and minima ordinates, if any ; 
Asymptotes and singular points, if any. 

It is useful to remember also that an infinite branch is 
convex toward its asymptote. 

EXAMPLES. 
1. Trace the cubical parabola d-ij = x^. 



Fio. 1, 

The origin is a centre of syniniotry and a \io\\\{ of infloxiotu to the 
right of which the curve i.s coiu'ave upward. The iutiuito brauchos 
are in the first aud the third (juadraut. ^^■lleu .r = ± oc, = 7r/2. 

For the form of the curve, see lii;-. I. 



138 



DIFFERENTIAL CALCULUS. 




Fig. 2. 



2. Trace tlie semi-cubical parabola (j?-''^y — x^''^^ or a?/^ = x^. 

The curve is symmetrical with respect to the 
X-axis, and lies to the right of the ?/-axis. 
The origin is a cusp of the firsli species. 
When X = 00 , =. ;r/2. 
For the form of the curve, see fig. 2. 

The curve 

a'*-i?/ = x» (1) 

is frequently called the parabola of the nth degree, 
n being greater than unity. 
When n = 2, (1) becomes 

ay = x2, (2) 

the locus of which is the common parabola with 
its axis on the ?/-axis. 

When n is an even integer or a fraction having 

an even numerator and an odd denominator, the 

general form of curve (1) is that of the common 

parabola (2). 

When n is an odd integer or a fraction with an odd numerator and 

an odd denominator, the general form of curve 

(1) is that in fig. 1. 

When n is a fraction having an odd numer- 
ator and an even denominator, the general form 
of (1) is that in fig. 2. 



3. Trace the cissoid of Diodes, 
y^ = x^/ (2 a — x). 

Y The X-axis is an axis of symmetry. The 
curve passes through (a, a) and (a, —a). It 
lies between x = and the asymptote x = 2 a. 

The origin is a cusp of the first species. (See 
fig. 3.) 

To construct the cissoid geometrically, draw 
any line OB from to XR, and take 

RP = OS ; 
then P will be a point on the cissoid. 




R 



Fig. 3. 



CURVE TRACING. 



139 




4. Trace the curve y^ = 2 ax^ — x^. 

y = — X + 2a/3 
is an asymptote. 

(2 a, 0) is a point of inflexion, 
to the right of which the curve 
is concave upward. Hence, 
the infinite branch in the fourth 
quadrant lies above the asymp- 
tote, and the one in the second 
below it. 

When X = 4a/3, 

?/ = 2aV4/3, 

a maximum ordinate. 

Fig 4 
The origin is a cusp of the 

first species, the ?/-axis being tangent to both branches. 

5. Trace the curve a'^y^ = a^x^ — x^. 

Each axis is an axis of sym- 
metry. The origin is an oscu- 
lating point of the first species. 

The curve is enclosed by the 
tangents 
x=±a,y= ±2aV3/9. 

When X = ± aVO/3, 

y = 2 a V3/0, a max imum. 

When X — ± a v 27 — 3 V33/(), (x, y) is a point of infle: 
fig. 5.) 

6. Trace the folium of Descartes y^ — o ((x// + X'> = 0. 

y = — X — a is an asymptote. 

The infinite branches are concjive 
upward, and hence lie above this 
asymptote. 

dy/dx = at (0, 0) 

and (rtV2, a-\^); 

dy/dx = op at (0, 0) 

3/— «/■ 

and (<« v4, (iv2). 

This indicates a double point of 
int(M-st>ction at the origin and a li>op 
tangent to Mio axes and the liu(>s 

y =■ (I VI and x = a^ \. (See tig. (5.) 



(See fig. 4.) 




Fig. 5. 



(See 




140 



DLFFERE^^TIAL CALCULUS. 



7. Trace the stroplioid x {:c- + y-) + a {x- — y-) = 0. (See fig. 7.) 

8. Trace the hypocycloid of four cusps x-^'^ + y'-''^ = a-^'^. 

Each axis is an axis of symmetry. 

The limits of the locus are a: = ± a, y = zna. 

dy/dx = - {y/xyri = o, at (- a, 0) or (a, 0), 
= ap, at (0, — a) or (0, a); 
hence, there is a cusp of the first species at each of these four points. 

d^y/dx- = a^^y3xi^-y^^^: 
hence, the curve is concave upward in the fii'st and second quadrants. 
(See fig. 8.) 





Fig. 



Fig. 



In order to examine this curve for cusps by the method of § 152, 
it would be necessary first to rationa.lize its eciuation. 

This curve is traced by a point, P, in the circle PB (fig. 8) as it rolls 
within the fixed circle XBYX', whose radius is f otir times that of the 
circle PB. 

9. Trace the evolute of the ellipse, 
(r/:c)i/3 + (6^)^/3 = (a%^)2/3^ 

where a and b denote the semi-axes, 
and e the excentricity of the ellipse. 

Each axis is an axis of sym- 
metry. The curve is enclosed by 
the lines x = ± ae-. y = ^ a-e-/b. 

dy/dx = — {a^y/W-x)^'^ - 0, 
at (— ae2, 0) or (ae^, 0); 

dy/dx = ap, 
at (0, — o:^e-/b) or (0, a^e^/b). 




CURVE TRACING. 



141 



Hence, there is a cusp of the first species at each of these four 
points. 

The curve is concave upward, in the first and second quadrants. 
(See fig. 9.) 

10. Trace the conchoid x^^y- — (6- — ?/"-) (a + yy-. 

11. Trace the witch of Agnesi (x'^ + 4 a^) ?/ — 8 a^. 

12. Trace the curve x^ + y^ — gZ^ 



Tt/'2,, cos-i(— 5/a), 
h, 0; 



13. Trace the polar curve p = a cos d + h. 
When ^ = 0, 7r/4, 

p = a + h, V2a/2 + 6, 

when ^ = 3 7r/4, tt, 

p=— V2a/2 + 6, — a + 6, 

These values of p will recur in 
reverse order as d increases from 7t 
to 2 ;jr ; hence, the locus is symmet- 
rical with respect to the polar axis. 

Locating these points for a = 3 and 
6 = 2, we obtain the curve in fig. 10. 

When a >> 6, the point is a double :fG 
point of intersection, as in the figure. 

When a = 6, is a cusp. 

When a <; 6, there is a point of 
inflexion at B and at S^ and is not 
on the curve. 

When a = b, this curve is called 
the cardioid. 

When a = 2 1), it is called the lima(^on. 



14. Trace the curve p = a sin 3 6. 

p = a, a maximum, when sin 3 <? = 1; 
that is, when = 7r/(), 5 7r/(), • • •, 

p = — a, a mininuim, when sin 'AO ^ — 1 ; 
that is, wluMi = tt /2 , 1 7r/(\, • • •. 

WluMu; - 0, 7r/(), 7r/;>, 7T/2, 2;r/3, i)z/CK tt, - • \ 
P ~ 0, (/, 0, — a, 0, a, 0, • • •. 

The i'urvi> consists of t/nrv equal loops. (See fig. 11.) 




Fig. 10. 



142 



DIFFERENTIAL CALCULUS. 



15. Trace the curve p = a sin 2 ^. 

The curve consists of four equal loops. (See fig. 12.) 
The locus of p = a sin ^ is a circle, a curve of one loop. 

From the number of loops when n = 1, 2, 3, we infer that the 
locus oi p = a sin nd consists of n loops when n is odd, and 2 n loops 
when n is even. 




Fig. 11. 



Fig. 12. 



Fig. 13. 



16, Trace the lemiiiscate p2 = ^'^ cos 2 d. 

The lemniscate is the locus of the 
intersection of a tangent to the equi- 
lateral hyperbola and a perpendicu- 
lar to the tangent from the origin. 
(See fig. 13.) 

17. Trace the curve p = a sin3(^/3). 
(See fig. 14.) 




Fig. 14. 



18. Trace the curve p= a cos 6 cos 2 d. 



19. Trace the logarithmic spiral p = e«^. 



PART II. INTEGRAL CALCULUS. 



CHAPTER I. 
STANDARD FORMS. DIRECT INTEGRATION. 

156. Integration. Having given the differential of a func- 
tion, inteAjTcition is the operation of finding the function. In 
other words, having given the ratio of the rate of a function 
to that of its variable, integration is the operation of finding 
the function. 

A function is called an integral of its differential. 
Thus, fx is an integral of fix) dx. 

157. The general or indefinite integral of any differential 
(^ (x) dx is the most general function whose differential is 

<l>{x)dx. The sign o£ indeji7iite iiitegrat 1071 is |. Tlius. | in 

the expression | cf)(x)dx denotes the operation of iinding the 

indefinite integral of cf>(x)dx, while the whoh^ expression 
denotes the indefinite integral itsc^lf. 

For oxaiuplo, if C is a general constant, .r-'^ + C is the most general 
function whose differential is '^x-dx ; that is, 

C-Ax-dx X'^ + C. 

x^ x^ — 1, X'^ + 8, • • • are parficiiliir inle_i;rals of ;>.r-(/.r. 

Tlie signs d and I indirati^ inrcrst- oporal ions, and, in gen- 
oral, mnilrali/A^ each other. 



144 INTEGRAL CALCULUS. 

Thus, d i ^(x)dx = <j) (x) dx ; 

jd{fx+C)=fx+C', 
^dfx=fx+ C, 



aud 
but 



where C denotes a general constant called the constant of 
integration. 

Notation. In the following pages we shall use fx-\-Cio 
denote the general or indefinite integral of the differential 
<ji (x) dx. 

158. Elementary principles. 

(i) r<t> W du = fu + C, if dfu = <|)(u) du. 

This principle affords the simplest proof of formulas for 
indeiinite integration. 



W-+1 

iC'du 



Thus, I ie'du = — 7—: -\- C, since d — -— 

J n-\-l n-\-l 

1 — = log 11 + C, since d (log u) = — ■ ; 



u ^ ' u 

, h C, since d - — 

log a log a 



I d"-du = :^ h C, since cZ^— — = a^du. 



In like manner all the formulas in § 159 can be proved. 

(ii) A constant factor can he transposed from one side of the 
sign of integration to the other without changing the value of 
the integral. 

For if a and c denote any constants, we have 

adij = d (ay -\- ac) ; 

. ■ . I adij = a(j/ -\- c) = a I d(y -\- c) = a I dy. 



STANDARD FORMS. 145 

(iii) The integral of a 'polynomial is equal to the sum of the 
integrals of its several terms. 

For du -^ dy -\- dz = d{ii -{- y -\- z -\r C) ; 

.-. i^ {du + dy ^- dz) = u + y + z + C 

= I die -\- I dy -\- i dz. 
(iv) Co= C, since dC = 0. . 

159. Standard forms and formulas. We give below a 
list of standard integrable forms. To integrate a differential 
directly, we reduce it to some one of these standard forms, 
and apply the formula. The most of these formulas are 
easily obtained by reversing the formulas for differentiation, 
and each is readily proved by (i) of § 158. The list will be 
gradually extended, and a supplementary list given later. 

u'^du = — -— + C, where n is not — 1. [11 

7l-\-l ' ■- -■ 

/^^iog..+ a [2] 

fa^du = :r^^-^ a [3] 

J log a "- -■ 

Ce''du = e''-\- C. [4] 

j sin udu = — cos u + C, or vers u + C [5] 

I cos udu = sin u + C, or — covers u + C. [(>] 

j sed^udu^tan u -f C. [7] 

I Gsch( du = — cot u -h C. [8] 



146 INTEGRAL CALCULUS. 



I sec u tan udu = sec u + C. [9] 

I CSC ti cot tcdic = — CSC ti + C. [10] 

I tan udu = log sec ti + (7. [11] 

j cot udu = log sin u + (7. [12] 

/u 

CSC wc^?^ = log tan ^ + C. [13] 

j sec udu = log tan ( o + T ) + ^- [14:] 

f-^^, = i tan-i !f + (7 or - i cot"^ - + C. [15] 

-^ -2 = — log— -— + C, or ^-log^-^H- C. [16] 

-== = sm-i - + C, or - cos-i - + C. [17] 
V a^ — ?r (^ <^ 

r r^,=iog(u+v^i^^^)+c. [18] 

— , =- sec-1 - + C, or csc"^ - + C [19] 

ic^tc^ — a^ a a a a 

f-^=£^= = vers-i - + C, or - covers-^ - + C. [20] 



Of these twenty forms only the first three are really /i^?ic?a- 
mental; for by proper substitutions each of the others can be 
reduced to one of these three. 

In the applications of these formulas we shall omit the 
constant of integration C, as it can readily be added when 
necessary. 



DIRECT INTEGRATION. 147 

160. The variable parts of the indefinite integrals of the 
same or equal differentials are equal or differ by a constant. 

For if the differentials of two functions are equal, the rates 
of these functions will be equal ; hence, the functions will be 
equal or differ by a constant. 

For example, in formula [5] vers u and — cos u differ by 1 ; and in 
[17] sin-i(u/a) and — cos-i(w/a) differ by 7r/2. 

161. Formulas [1] and [2] may be stated in words as 
follows : 

The integral of a variable base with any constant exponent 
{except — 1) into the differential of the base is the variable base 
with its exponent increased by 1, divided by the new exponent. 

The integral of a fraction whose numerator is the differential 
of its denominator is the Naperian logarithm of the denominator. 

Formula [1] fails to give ^finite result when ti = — 1 ; but 
formula [2] provides for this case. 

Formula [2] gives a real result only when u is positive. 

EXAMPLES. 
By § 158 and formula [1] or [2], find 



1. I a'jif'd£ = a I x^dx = ax'/7. 



The meaning of this result is that ax"^ / 7 + C is the most general 
function which changes ax^ times as fast as .r, or that ?/ = ax' /I + C 
is the equation of the family of curves whose slope is ax^. 

►4 dx 
2. I —7- =4 I x-^dx = -X 



/4 dx /» 

Here u =^ x, n = — 5, and h + I = — 4. 

3. j (X- — a-)3 /•-' xdx = 7y j {'I'- — tt-Y^' ■ - '^dx - (j-- - (j-)^' /-V^- 

Here n = 3/2, u — x- — a- ; .-. du = 2 xdx. 

Hence, we introduce the faclin- 2 bot\n-c xdx and write its recip- 
rocal 1/2 on tlio lol't o( the sign of integration. 



148 INTEGRAL CALCULUS. 



4. j a(x + l)2(^x = a(x+ 1)3/3. 

5. Cilax^/^ — lbx^/^)dx = ^a ( x^/^dx — ^b Cx^^-^dx 

-aLx'i-^ — hx^i-\ 

/dx r- /•/l^ 6 \ 

V5 = '"^- 8-jfe-x.)'^- 

Wlien the integral is not given, the result obtained by integration 
should be verified by principle (i) of § 158. 

10. j 6 (6 axP- + 8 6x3)5 / 3 (2 ax + 4 6x2) ^r^ 

= ^ 1(6 ax2 + 8 6x3)5/3 (12 ax + 24 6x2) (^^^ 
= 6(6ax2 + 8 6x3)8/3/16. 

11. j [a (ax + 6x2)1/3 ^o; + 2 6 (ax + 6x2)i /3 a:cZx] 

= I (ax + 6x2)1/3 (a + 2 6x) dx = f (ax + 6x2)4/3. 

12. r(2a + 36x)3cZx. 13. (^(1 + 9x/4)i/2dx. 

/x'^-idJx 1 /*/ , ^ N ^ -1^ (a + 6x«)i-'« 

: — , ^ . = 7- I (a + 6x")-»' 6nx«-i dx = \ ,, — ^— — • 
(a + 6x«)™ bnj^ ' bn (1 — m) 

17. CV2pxdx. 19. r(2x4-3x2 + l)i/2(x3-3x/4)(Zx. 

_- /» dx ^d (x — a) , , , 



DIRECT INTEGRATION. 149 



Here u = a -{- 6x» ; .-. dw = n&x'«— i dx. 



23 



10x3+ 15' '^- J ae + &x3 

X + 1 , «^ /^ 5 6xcix 



/ x + 1 /* 5 oxrtx 

x2 + 2x'^'^- ''^- J 8a-66x2* 

Expand (2 a — x^)^ and then integrate. 

27. r (6 - X2)3 Xl /2 dX = I 63x3/2 _ 6 52^7 /2 + ^6_ ^^H /2 _ _2^ ^l^ /2. 



5x^dx «o. /*., X ^33 (lo2:x)'" + i 

;•• 30. I (logX^"' — ^ ■ ' 

dx „, /• c?x 



28- js^f + T- 30. j (log.) ^ ^^^^^ 



29 



J(logx)3f • 31, J-||- = ,og(IogX). 

= — I -^ -' = — log (a — x). 

a — x J a — X ® ^ "^ 

34. r (1 + X) (1 - x2) x(Zx = X V-^ - a^V ^ + .^••'' / '> - -r'^ / 5. 

35. C'^^^dx = Hx + 1 + -ZTt)'^-^' = l'"^- + •^' + - ^'"'^i'^' - 1) 



32, 



33 



36. f-i"^-^ (/x = X + loo;(nx - 1 )-/->. 
^7 o .t: — 1 ^ ^ ' 



150 INTEGRAL CALCULUS. 

//oS/S — 0.2/3)3/4 3 /» / 2 \ 

^ Js ^ cZa::^-|J(a-^/-3-x2/3)3/4(^_|^-i/3^^-) 

= — f (a2/-3— x2/3)7/4. 



^/^^^" 



2a:-2 - l)-3/2 (- 2 a2x-3 dx) 



_ (a2ar-2-i)-i/2 _ 



a^ a2Va2-x2 

Sometimes, as above, we may transfer a variable factor from the 
base to the differential factor, or vice versa, and thus make the differ- 
ential factor the differential of the base. 

dx r ,. , n .. . ,. o , Vx2~+a2 



4L r . = f(i-\- a2x-2)-i/2 

J x2vx2 + a2 J 



a^x 



An i tZx .. /^ Va2-.x2 (a2- 3^2)3/ 2 

42. a 44. <— ^^^-- ^ 3^2^ 



, • 44. I I 

x2Va2 — x2 J ^ 

C- ^^^^TT7-,= r(2ax-i-l)-3/2x-2dx^ , "^ 



,^ , dx ^^ /^Vx2-a2 3^2-0,2)3/2 

43. I T-r^ — ^7^-77, • 45. I 7 dx = -5^ — —-^ — 

x* 8a2x3 



, V2 ax — x2 , _ r dx V2ax — x2 



/^^-- - /; 



X V2 ax — X- ^^ 

By § 158 and one or more of the formulas [1] to [4], find 

/I /» a*^ 

4^ 4 log a 

Here u in [3] is 4 x ; .-. du = 4 dx. 

50. Te^/'' c7x = n Ce-"/» — = ne^i^. 

51. C a&^dx. 52. j ca'^^'dx. 

53. flog c ■ c^*x3dx = ^-^ fc^^ • 4 x3dx = c^V4. 



DIRECT INTEGRATION. 151 



54 i (ci^''^ — b™^) dx 

J n log a m log 6 

56. r(2e3^-a)V2e3x(^x = (2e3^- a)5/V15 



/It!- - / 



log 6 + logc 



162. To obtain formulas [11], [12], [13], and {l^from [2]. 

//'tan w sec udu 
tan ?^tm = I ■ 



sec u 
log sec u + C [11] 

cos tidu 



/cot ?t^w = I 
J sm ?t 

= log sin u + a [12] 

I CSC udu e: I -: 

J J sm ?i 

"J 2 siu(/r/2)cos(;^/2) 

"J tan(?//2) 

-log tan (7^/2)+ C. [13] 

i sec udii _ I CSC (/^ + 7r/'2)du 

= log tan (/^/2 + IT/ 4) + a [14] 



152 INTEGRAL CALCULUS. 

EXAMPLES. 
By § 158 and one or more of the formulas [1] to [14], find 

Here u in [5] is mx ; hence, du = mdx. 

2. I sec'^ mxdx = m—'^ t2bn mx. 5. | cosxsinxdx. 

3. I sin^rc cosa:dx = sin5cc/5. 6. | 7 sec^ x'^ • xdx. 

4. j sec^x^ -x^dx^ tancc^/S. 7. | cos (a + bx) dx. 

8. rcos43^sin3^(Z^= -cos53^/15. 

9. I sin3 2 x cos 2 xdx. 12. j e^osa; ginxcix. 

10. j 5 sec 3 X tan 3 xdx. 13. j e^ ^inx cos xdx. 

11. j4csc«.cotax... 14. J„-c.,ec^,,,,. 

/•sin xfix 1 /• — 6 sin xdx 1 ^ . , , 

15. I — 7—. = — T I — ^7 = — - log (a + cosx). 

J a + 6cosx 6^ « + 6 cosx 6 °^ ' 

16. I ^^-^-j — ^^ = log {ad + sm 6). 
J ad + smd ^ ' 

17. I sec u CSC Mc^M = | — = log tan u. 

J J tan It 

18. Jsec^. csc^«.« = J(sec^. + cse^u).. = tan« -cot. 

19. Csm^udu= j (1/2 — cos2M/2)cZit = w/2— sin2w/4. 



DIRECT INTEGRATION. 153 

20. Jcos=«.« = »/2 + .„2«/4. 

21. j tan^wdit = tanw — u. 

22. I (sec u — tan u^ du = 2 (tan u — sec w) — u. 

oo /'I ~ sin w , /'(I — sin u)'^ , o /+ x 

23. I -— — : — du = I -^ — - du = 2 (tan u — sec u) — m. 

^ 1 + sm u I cos^ u 

24. I — : du — { sec 2 udu = - log tan { u + — )• 

J cot w — tan u J , 2 \ 4 / 

163. To obtain formulus [16] (xwc? [18]/ro??i [2]. 

I l l 11 

zi^ — a^~2ai^ — a 2 au -{- a 

/ du _ 1 r du 1 r du 

u^ — a^~2 aJ u -— a 2 a J u -\- a 

1 , u — a ^ ^ ,^ ^ 

/^w _ 1 r — c?7i 1 r r/?^ 
u"^ — a?~ 2 a J a — u 2 a J a -\- a 

We use the form in (1) or (2) according as u — a or a — u is 
positive ; that is, in each case we take the form which is real. 
To obtain formuhx [18], h^t 

Vvr ± rr = r: — u ; (1) 



or 



then 



a- = z- — ^ u 



> ,/~ 



.'.(p- u)dz = T:du\ 



dr: du du 

or — = = — z__=- bv (0 

/ , = I — = l<-^g ~ = log" ( ii + V«=^ zb «l bv ( 1 ") 



154 INTEGRAL CALCULUS. 



EXAMPLES. 



5(6x)__ _ 1_ _j &^ 
■i- c^ be c 



bx — c 

bx + c 



n dx _ 1 /* d( 

■ J b-^x^ -i-c-^~ bj {bxy 

r dx ^ir d(bx) ^J_i ' 
J b'^x^-c-^ bj (bxf — d^ 2 be ^^ 

/* dx _ /^ dx _ 1 _^ x + 2 

"*• J x2 + 4x + 9~J (x + 2)2 + 5~ V5 V5 ' 

r dx _ /» dx _ 1 X + 2 — Vs 

• J x2 + 4x+l~J (x + 2)2-3~2V3 ''"X + 2+V3 

/* xdx _ 1 _i^ 17 /* dx 

■ J a4 + x4~2a2 ^^ a2' J 16x2 + 9* 

/^ x^dx _ 1 ^ x^ — 1 f* dx 

6- J ^^—r T-6^^-^M^" ^' J 9^^^i' 



•■/^ 



10. 



dx 1_ bx — c _ _J_ 6x + c 

-6-2x2- 2 6c °^6x + c~26c °^6x-c 

dx ^ /* 2 adx 



/ dx _ /* 2adx 

ax2 + 6x + c "J (2 ax + 6)2 + 4 ac — 



2 , 2 ax +6 ... 

= =tan-i > (1) 

V4 ac — 62 V 4 ac — 62 

1 , 2 ax + 6 - V62 - 4 ac 

or . log , • (2) 

V62 — 4 ac 2 ax + 6 + v 62 — 4 ac 

We use the form in (1) or (2) according as 4 ac >> or < 62 ; that 
is, in any given case we take the form which is real. 

r dx _ r Sdx _ 1 _^ 3x + 2 

• J 3x2 + 4x + 7~J {3x + 2)2 + 17~ Vl7 ^° Vl7 * 



12 f ^^ + ^ dx - fi^xj^ii^ 4. o r 

^^' J x2 + 4x+5'*''~ J x2 + 4x+5^^J 



dx 



(X + 2)2+1 
= log(x2 + 4x + 5) + 3 tan-i (x + 2). 



DIRECT INTEGRATION. 155 

(1) 



+ 5 



r (8 x + 2)dx _ 3 /* (4 X + 4) da; _ r dx 
J 2a;-^ + 4x + 5~4j 2x2 + 4x + 5 J 2x2 + 4x 

=:7log(2x2 + 4x + 5)- -^tan-1^^^- 

In like manner any differential of the form ■ /, , \ can be 
. . -, ax- + ox + c 

mtegrated. 

Note that the numerator of the first of the two fractions in the 

second member of (1) is the differential of its denominator. 

14- Cz^^^-rZT dx = l log (x2 + 4 X + 5) - tan-i (x + 2). 



x2 + 4 X + 5 2 



■'/ 



2x-3 , 1. ..„,.. 3 . ,xV5 



dx = - log (5 x2 + 4) j=. tan- 



5x2 + 4 5 ^^ ' 2V5 2 



16. fz^^^^n-.=^og{x-hl)+ ' 



X2 + 2X+1 °' 'x+1 

bdx 1 . ,bx 



17. r^M==ir 

J Va2c2-62x2 bj 



■V{acy^- (6x)2 i> 



18. r , _ ^'^ = = J log {hx + V6--^x2 ± a2c2) 



19, 



V6'-ix2 ± a'-c'-^ & 

dix /• (f X . , 2 .r + 1 



/ dx _ r 

Vl — X — x2 J 



V5/4 -(X + 1/2)- Vo 

20. r . ^^'^ = -^ log (X Vrt + ^^^r- - ?>) . 
,/ Vax"- — & v« 

21. C—=J^= = log (X + a +^V- + 2(T.r). 
. / V x"- + 2 a.(: 



22. 



23. 



rxi/2c2.x 1 r ^x>''^dx 1 . , /x3 

- =: - I — = - sin— Hi— • 

V8-4.<;'> '>J V(2V^)--C^^-"'/-)- '• ^- 

r (?x _ _1_ r 2 adx 

V(xx2 4 6x + c VaJ V(2 ax + 6)- + 4 ac — b'^ 



= — ^ loi;- (2 ax + /> + 2 n a \ ax'^ + bx + c). 

^'a 



156 INTEGRAL CALCULUS. 

dx 1 /* 2adx 



24. 



25. 



ox _1_ /» 

V— ax- + hx + c ^aj 



+ bx + c VaJ V4 ac + 6-^ — (2 ax — 6)^ 



1 . , 2ax — 6 
= — ;= siir 



V^ V4 ac + 62 

/ (?x _ _1_ ^ 4dx _ J_ • _ 1 4x — 3 

V4 + 3x-2x2 V2J V41-(4x-3)2 V2^^^ Vil 

fZx 



27. 



/ ^'^ ^rr = 4= log (4 X + 3 + 2 V2 V2x2 + 3x + 4) . 
V2 x2 + 3 X + 4 V2 

r , ^^ 29. r—=^==. 

J V3x-x2-2 J V2 x2 + 2 X + 3 



28. f ^ ^^ . 30. f_^|^. 

J Vi + x2 + X J Va* - x* 

„- /* dx _ /* cdx }_ _iCx 

J xVc2x2-a262 J ex V(cx)2 - (a6)2 a& «& 

32 r ^'^ 33 r 5dx 

J X V6-2x2 - a2 ' J xV3x2-5 

34 C dx _ r V7dx 1 _ ,xV7 

»/ V7 x* — 5 x2 •/ xVT- 



V7 x2 - 5 V5 V5 

— cdx 1 , ex 

, - = - covers- 1 — 

V8 • ex - (cx)2 c 4 



_^ /* — dx 1 /• — cdx 1 , ex 

35. I ,. = - ! ■ , = - covers- 1 

J V8 ex - e2x2 cj ■ 

36. r-=^=- 

J V2a6x-62x2 
gg /- (2x + 3)dx ^ /* (2x + l)dx ^ ^ /- dx 

* J Vx2 + X + 1 J Vx2 + X + 1 J Vx2 + X + 1 

= 2 Vx2 + X + 1 + 2 log (2 X + 1 + 2 Vu;2+x + l). 

In like manner any differential of the form /• can 

be integrated. Vax2 + 2 6x + c 

Note that the numerator of the first of the two fractions in the 
second member of (1) is the differential of the base in the denomi- 
nator. 



DIRECT INTEGRATION. 157 



r* —xdx _ r c — 2x — c 

J Vcx — x'^ J 2Vcx — x2 

/ o c , 2 X 

= Vex — X- — - vers— i — 
2 c 



/^ (x2 — a^)i/2c?x _ /* (x2 — a2)dx 
'J »^ J xVx2 — a2 

X dx /• aMx 



/ xdx _ n __oMx__ 
Vx2 — a2 J xVx^ — a2 

= Vx'-^ — a^ — a sec- ^ - • 



41. 



/ , dx = sin— 1 X — Vl — x^. 

VI -X 



42. f f "^^ dx = sec- 1 - + log (x + Vx2 - a'^). 
•^^ X Vx — a ^ 



CHAPTER II. 
DEFINITE INTEGRALS. APPLICATIONS. 

164. Definite and corrected integrals. Let h> a-, then 
the increment produced in the indefinite integral fx-\-Chj 
the increase of x from a to b is 

fb+C-(fa+C)=fb-fa. 

This increment of the indefinite integral of (fi(x^dx is called 
"the definite integral of (fi(x)dx between the limits a and ^," 

and is denoted by | cf)(x)dx. 

Hence, f </,(x) dx =fb -fa. (1) 

c/ a 

Xb /*h 

in the expression I <^(x)dx denotes the 

operation of finding the increment of the indefinite integral 
of (fi(x)dx from x = a to x = b. b is called the 'upper' or 
' superior ' limit, and a the ' lower ' or ' inferior ' limit. 

The limits a and b must be so chosen that (ftx will be finite, 
continuous, and of the same quality, from x = a to x = b. 

The expression/^ —fa is often denoted hjfx . 

If, in (1), we make the upper limit variable and put x in 
the place of b, we obtain 

I cl>(x)dx=fx-fa, OT fx\ , (2) 

•/a _Ja 

which is called "the corrected integral of (^(x)dx.''^ 

By I cf>(x)dx is meant the limit of I <ji(x)dx when x = co. 



DEFINITE INTEGRALS. 159 



EXAMPLES. 



X4:-[X X* _ 1 

4 4 



2. Cnx dx = ~ x^l ^= ^ (62 - a"^). 

Jo 1+^' 2 J^ 4 

6x3dx = 24. 10. I x"dx = - 

•) a 



X2). 



«/o 12 .A) -y/^r — y 

7. I -v~^ — ; = T— • 12. I — (y- — 62)4 dij — —— — - • 
Jy a2 + x2 4 a J_5a4 ^lj«'* 

8. I -—= = -• 13. I e-"'^dx = -- 

Jo ^(f^ — X2 ^ Jo « 

9. I ■ r- =V2 — 1. 14. I siii'^x cos\r (/.r = — • 

Jo ^0^2^ J^, 12 

15. I = sin~ ' — ^r- I =8111-^ — — siii-'-7r- 

Jo V 5 — 2 X — x2 V() J VO VO 

^j o + 2 X + x2 V2 V V2 / 

J-»x T-e — 1 -L at— 1 1 

, x« + e- c- ; ^ 



160 INTEGRAL CALCULUS. 

165. Corresponding definite or corrected integrals of equal 
differentials are equal. 

For the corresponding increments of variables which change 
at equal rates are equal. 



G .3a; -1 , „ , . . 16 



Ex. r 7S__oZ2 ^^ = 3 log 3 4- 



Let z = x — Z] then a; = 2 + 3, (Zx = d2, and 

3x-l ^ 32 + 8^ 
ax = ; — dz. 



(X - 3)2 ^ 

When X == 4, 2 = 1, and when x = 6, 2 = 3 ; 

•••i (¥^^2^^-j^ -^^^ ^ §165 

==[31og2-8/z]^ 

= 3 log 3 + 16/3. 

This example and those which follow illustrate also how 
the introduction of a new variable often simplifies a given 
differential and renders it directly integrable. 

EXAMPLES. 

1. I _ dx = I ^ (iz = log 3 + 4, where 2 = x — 3. 

„ r'^ xMx r^ { z-\)dz 3 .If- , ox , Q , 1 

2. j^ (^2 + i)./3 -j^ ^^^ = -(V5 + 3), where. = x2 + L 

/*« dx _ _ 1 r^__^^__ 

Ji xVa2_x2~ aj„ V.s-i 

= log (a + Va2 — 1) /a, where z = a/x. 

/•i__e^^dx__ /^g + i (z — l)dz 
• J„ (ex + 1)1/4 -J^ 

= 2T [(3 e - 4)(e + 1)3/4 + v^]^ wiiere z = e^ + 1. 



1, a + Va2 + 1 



/•« dx 1 - a + Va2 + 

5. I — ■ = - log ^ 

Ji X Va2 + x2 « 1 + V2 

In example 5 put x = a/z, in example 6 put ^ + a = 2. 

X& — « cos ^d^ 
-— — - = (b + a) cos a — sin a log (cos 6 /cos a). 
2„ cos(^ + a) 



AREAS OF CURVES. 



161 



<^(x)dx, I <^(x)dx, I </)(x)dx. 

Let SPQ be the locus of y = <^x. 

Conceive the variable ordinate y or NQ to trace the area 
between the ic-axis and y = cf>x SiS the point (x, y) moves along 
the curve to the right. 

Let A denote the area bounded by the cc-axis, y = <fix, some 
undetermined fixed ordinate, as ES or H'S', and the moving 
ordinate NQ. 

Let dx = NN'; then dA = NQQ'N'; 

.'. dA = ydx = (f>(x)dx. 



.'. I ydx = I <l>(x)dx = A. 



(1) 

(2) 




Let OM = a, and OJf ' = h ; then the increment produced 
in A by the increase of x from a to ^ is MPP'M') 



J ydx = i cf>(x)dx 

_ r area bounded by the ^--axis, 1 
\ y = <^.r, X = «, and a" = h. J 



(3) 



Similarly, if OM = ^? and OjY = .r, we have 

area bounded by the .r-axis, ^ 

ij = (jf).r, X = a, and the orilinate //. I 



= { 



n 



In (2), A is indotiMMiiinati^ so Ioiilc as ilu^ ti\od oriliiiate /I'S. 
or E'S\ is iuiU>t(M'niinat(\ 



162 INTEGRAL CALCULUS. 

From (1), by Cor. 1 of § 12, we know that the areas in (3) 
and (4) will be positive or negative according as (f>x is posi- 
tive or negative from x = a to x = b. 

Hence, if a curve crosses the ic-axis the area above, and the 
area below, this axis must be obtained separately. 

Eor example, to find the area bounded hjy = (px, the x-axis, and M'P% 
we find the areas Bf'S'K and KFP'M' separately and take their arith- 
metical sum. 

From the geometrical meaning of an integral it follows that 
<f>(x)dx has an integral whenever cfix is continuous. 

Ex. Give the geometric meaning of the definite integral in each of the 
examples on page 159. 

EXAMPLES. 

1. Find the area bounded by the x-axis and y = x^ -\- ax^. 

The curve cuts the x-axis at (— a, 0) and (0, 0); hence, the limits 
are — a and 0. 
Here dA = ydx = {x^ + ax^) dx ; 

: r (x-3 + ax2) dx = rxV4 + axVsl by (3) 

= aV12. 
In each problem the reader should first gain a clear idea of the 
boundary of the figure whose area is required. 

2. Find the area bounded by the x-axis, the parabola x^ -f- 4 ?/ = 0, 
and the line X = 4. Ans. 16/3. 

Since the area lies below the x-axis, the formula gives a negative 
expression for it. 

3. Find the area bounded by the x-axis, the curve y = x^, and the 
lines x= —2 and x = 2. ^^s. 8. 

Here we find the area below, and the area above, the x-axis sepa- 
rately, and take their arithmetical sum. 

4. Find the area bounded by the curve y = e^, the x-axis, the ?/-axis, 
and the line x = h. ^^s_ qA — i. 



.-. area 



AREAS OF CURVES. 163 

5. Show that the area bounded hy y = x — x^ and y — is 1/2. 

6. Eind the area bounded by the parabola y'^ = 4_pic and any chord 
perpendicular to its axis. Ans. ^.xy /?j. 

Here the area required is twice the area bounded by the x-axis, 
y — 2 pi/ 2 a;!/ 2^ a^nf^ ^j^e ordinate y. 

.-. area = 2 r^2pi/2xi/2(^ic = (2/3)2x2/; by (4) 

that is, the area is 2/3 the circumscribed rectangle. 

7. Find tlie area bounded by the witch ?/(x2 + 4a2) = Sa^ and its 
asymptote y = 0. 

Area = 2 I ., . , ^ = 8 a^ tan- ^ r— \ =4 7ta^. 
J^ x2 + 4a2 2aJo 

Here the area between the curve and its asymptote is finite. 

8. Find the area bounded by the hyperbola x?/ = 1, its asymptote 
y = 0, and the lines x = 1 and x = a. Ans. loo- a. 

When a = 00, log a = oo ; hence, the area between the hyperbola 
and an asymptote is infinite. 

9. Find the area bounded by the curve y = cos x, the x-axis, the 
?/-axis, and x = 7t. Ans. 2. 

10. Show that the area bounded by the curve y = tan .r, the x-axis, 
and the asymptote x = 7t/2 is infinite. 

167. Formulas for accelerated motion. Lot t denote a 
portion of time, s the distance traversed by a moving- body. 
V the velocity, and a the acceleratii>n ; iluMi we have the fol- 
lowing formulas : 

(i) v=ds/df; .'.s= Cnlf, f= Cds/r. 
(ii) a = du/df; .'. >' = ( ^^dt, f -- i dr/a. 



164 



INTEGRAL CALCULUS. 



EXAMPLES. 

1. To find the fundamental formulas for uniformly accelerated motion. 

Let ^0 and Sq denote, respectively, the initial velocity and distance ; 
that is, the values of v and s when t = ; and let a' denote the con- 
stant acceleration. We then have 



V = Vo + I a'dt = a't + Uq, 



So + 



I 



vdt 



a'V^/2 + Vot + So. 



(1) 
(2) 



If u = and s = when i = 0, (1) and (2) become 



t — ^'28/ a\ v=^2a' 



(3) 
(4) 

The acceleration produced by gravity at the earth's surface is 
about 32.17 ft. a second, and is usually represented by g. Substitut- 
ing g for a^ in equalities (3) and (4), we obtain the four formulas for 
the free fall of bodies in a vacuum near the earth's surface, 

. A rifle ball is projected from in the direction OY with a velocity 
of c feet a second. Find its path, knowing that 
its velocity acquired in t seconds along the action 
line of gravity OX is gt feet a second. 

Let OX and OY be the co-ordinate axes ; 

dy/dt = c, dx/dt = gt; 




then 



••• y=\c 

c/O 



cdt — ct, 



x = gty2. (1) 



Eliminating t between equations (1), we 
have 

y2 = 2(f-x/g. 



Hence, the path of the ball is an arc of a parabola. 



3. A body starts from 0, and in t seconds its velocity in the direction 
of OX is 2 act^ and in the direction of OF it is a:-P- — c^ ; find its velocity 
along its path Onm, the distances in the direction of each axis and along 
the line of its path, and the equation of its path, the axes being rect- 
angular. 



ACCELERATED MOTION. 



165 



Let OX and OY be the axes, and let s denote the length of the 
path Onm ; then 



= -Aact; 



(1) 



. X = I 2actdt = act^ ; 
y= i {aH-^-c-^)dt = aHy3-cH; (2) 

.-. s = C'iaH'' + c2) dt = aV/Z + cH. (4) 



dt 




Eliminatmg t between (1) and (2), we obtain 



/ax o\ ^ 
which is the equation of the path Onm. 



(5) 



4. Given v =ft, to represent the time, the velocity, the distance, and 
the acceleration geometrically. 

Construct the locus of v =ft^ t being represented by abscissas and 
V by ordinates ; then the distance will be represented by the area 
between v =ft and the x-axis, and the acceleration by the lino rep- 
resenting dv when dt is represented by a unit length. 

5. A body is projected upwards with a velocity of 80 feet per second ; 
find in what time it will return to the place of starting. 

Ans. 5 seconds, nearly. 

6. From a balloon which is ascending at a uniform velocity of 20 feet 
per second two balls are dropped, one of them ;"» seconds before the other ; 
find how far apart they will be 6 seconds after tlie tii-st one was dropped. 

Ans. ;>*.>S feet, nearly. 



166 INTEGRAL CALCULUS. 

168. Change of limits. The following formulas are readily 
proved from the definition of a definite integral : 

cf>(x^ dx — —\ <^(x)dx ; (i) 

for each member =/^ — Z*^? if dfx = <^(x^ dx. 

cfi(x)dx= I cfi(x)dx -\- I <t)(x)dx; (ii) 

for the second member =/c — fa -\- fh — fc =fb —fa. 

Xa /»a 

<ji(x)dx= I <^(a — x)dx) (iii) 

for the second member = — I ^{a — x) d(a — x) 

Jo 

= -/(''-^)]"s/«-/0. 

Note. The Integral Calcukis was invented to obtain the areas of 
curvilinear figures, or for quadrature, as it is often called. 



CHAPTER III. 
INTEGRATION OF RATIONAL FRACTIONS. 

169. Decomposition of fractions. When the numerator of 
a rational fraction is not of a lower degree than its denomi- 
nator, the fraction, said to be improper, should be reduced to a 
mixed expression before integration. For example, 

^4 5 ^2 _ 4 

o , ,> ., 7, dx = (x — 2) dx + 3 , ,-■ ■■ dx. 

x^ + 2x^ — X — 2 ^ ^ x^ -^2x^ — x — 2 

The numerator of this new rational fraction is of a lower 
degree than its denominator. Such a fraction, called 2i> proper 
fraction, if not directly integrable, can be decomposed into 
partial real fractions which are integrable. These partial 
fractions will differ in form, according as the simplest real 
factors of the denominator of the given fraction are : 

I. Linear and unequal. 

II. Linear and equal. 

III. Quadratic and unequal. 

IV. Quadratic and equal. 

To present the subject in the simplest manner we solve 
below particular examples in each of the four cases. 

170. Case I. To each of the unequal linear factors of the 
denominator as x — a, there will correspond a partial fraction 
of the form A / (^x — a). 

The denominator x-' + x- — 2 x _x{x — 1) [x -\- 2). 



168 INTEGRAL CALCULUS. 

Hence, by addition of fractions we know that the given fraction is the 
sum of three fractions whose denominators are x, x — 1, and x + 2, 
respectively, and whose numerators do not involve x. 
We therefore assume 

2a: + 8 ^A B C 

X (X - 1) (X + 2) X X - 1 X + 2 ' ^^ 

in which A, B, and C are unknown constants. 
Clearing (1) of fractions, we obtain 

2x + 3 = ^(x-l)(x + 2) + 5(x + 2)x+ C(x-l)x (2) 

= {A + B+ C)x^-\-{A-^2B- C)x-2A. (3) 

Equating the. coefficients of like powers of x in (3), we have 

A + B+C = 0, A + 2B— C = 2, -2A = S. (4) 

Solving system (4), we obtain 

A = - 3/2, B = 5/3, C= - 1/6. 
Substituting these values in (1), we obtain the identity, 
2x + 3 3 _L 5 1 



x3 + x2 — 2 X 2 X 3 (X - 1) 6 (X + 2) ' 

(2 X + 3) dx _ _ 3 r*dx 5 /^ dx _ 1 /* dx 

+ 2 



/ (2 X + 3) (Zx _ 3 rdx 6 r dx 1 /»_ 
x-3 + x^ — 2x 2 J X 3 J X — 1 bJ X 

= - I logx + I log (X - 1) - 1 log (X + 2). 



The values of vl, 5, and C may be obtained directly from (2) as 
follows : 

Making x = 0, (2) becomes 3 = — 2 ^ ; .-. J. = — 3/2. 
Making x = 1, (2) becomes 5 = 35; .-.5 = 5/3. 

Making x = — 2, (2) becomes — 1 = 6 C ; .-. C = — 1 /6. 

EXAMPLES. 

^ r^^-i-. ,3- x-2 

1. I -r dx = X + 7 log —J • 

Jx2 — 4 4''x + 2 

^- f xf+^X^-ex ^^ ^ ^°^ ^'^''' (X -2)1/2 (X + 3)1/3]. 

„/»x — 1 , ./•5x + l, 
3. I ., , ^ r^f^a^- 4. I -^— -dx. 

Jx2 + 6x + 8 Jx2 + x — 2 



/^ (X2 + 1) dx ^1 ^ (X + 1)6 (X - 1)2 

J (2 X + 1) (x-2 - 1) ~ 6 °^ (2 X + 1)5 



RATIONAL FRACTIONS. 169 



6- I — z -. dx = — + — + 4 X + log / , ' • 

J x^ — 4 X 3 2 (X + 2)-5 

= -log(x2 - 6x + 7) + -^log 



8. 



2 2V2 X-3+V2 

/ xdx /* x^cZx 

(X - a) (X - 6) * ' J (X + 1) (x^ + X - 6) ' 



171. Case II. To r equal linear factors of the denominator 
as {x — by, there will correspond a series of r partial fractions 
A . B . . L 



of the form 



(x — hy {x-hy-^ x — b 

3 x2 — 7 X + 6 , ^ X — 2 

(X - 1)2 



Ex. 1. I — ^ --^ — dx = 3 log (x — 1) + 



Expressing the numerator in powers of (x — 1), we obtain 

3x2-7x4-6 = 3(x-l)2-(x-l) + 2; (1) 

for 3(x- 1)2 = 3x2 -()x + 3 
and — X + 3 = — (x — 1) + 2. 

Dividing both members of (1) by (x — 1)'\ we obtain 

3x2-7x4-0 _ _3 1__ 2 

(X - !)•' "^ X - 1 (X - 1)2 "^ (X - 1)'^ ' ^"^ 

/ 3x2 — 7x + () _ o r _i^ _ r _^-i* _ , , r_j^_ 
(X - 1)^' ' '' - \) X - T J (X -1)2 + -J (X - 1)=^ 

= 3 log (X - 1) + (X - 1)- I - (X - I)--'- 

From (2) we see that to resolve the given fraction into partial fractions 
by undetermined coellii'icnts, we should assume 

3x2-7x + _ A n _C_ 

(X - 1)« (X - 1)-' (X - 1)2 "^ X - r 

and proceed as in § 170. 



170 INTEGEAL CALCULUS. 






Here both Case I and Case II are involved ; hence, we assume 
1 ^ , 5 , C 



(2) 



(X-1)2(X+1) (X-l)2 X-1 X + 1 

Clearing (2) of fractions, equating the coefficients of the like powers 
of X, and solving the resulting system, we obtain A — 1/2, B = — 1/4, 
C = 1/4, Substituting these values in (2) and integrating, we obtain (1). 

EXAMPLES. 
10 



/Q 7- — 4 
^^^^dx = 31og(x + 2) + - 

^ /'2x2-3x + 4, ^, . o- 18X-41 
2- j^^3^)i-^-^ = 21og(x-3)-^^^^^. 

^ /-4x2--6x + 7^ 1, ,,^ , ,, , 9 25 

^- J (2x + 3)3 ^^ = 2^°S(^^ + ^) + 2(2x + 3)-4(2x + 3)2- 

p (2x-5)dx ^ 7 11 2(x+l) 7. 

J^ (X + 3) (X + 1)2 2 (X + 1) 4 ^ X + 3 4 

5- J% + 5x2 + 8x + 4 = ^ + ^°^^^ + l)-'- 

^ /•(2x3 + 7x2 + 6x + 2)dx . r . , ix/ ^ \^'^1 1 
6-j ^ x^+3xB + 2x2^ ^^"4^^^ + 'K^+'2) J-x- 

/* x2dx /* (x + 1) c^x 

J (X - 1)3 (X + 1) ' ' J (^ - 1)^ (^ + 2)2 * 

172. Case III. To each of the unequal quadratic factors of 
the denominator as x^-\-px + q, there will correspond a partial 

fraction of the form -^—, ; 

^ . /» (X2+1)C?X 1 ^ - X , 1 ^ 1 /T; /-,^ 

Ex. 1. { , ., I ox /o 9 J 1 X = — F tan-1 —= H p tan-i x V2. (1) 

J (x^ + 2)(2x2 + l) 3V2 V2 3V2 

By addition of fractions we know that the given fraction is the sum 
of two real fractions whose denominators are x2 + 2 and 2 x2 + 1 respec- 
tively, and whose numerators cannot be above the first degree in x. 



RATIONAL FRACTIONS. 171 



Hence, we assume 

x^ + 1 _ Ax-h B CX + D 

(x2 + 2) {2 x2 + 1) ~ x^ + 2 2 x^ + r 

Clearing (2) of fractions, we obtain 

x2 + 1 = (2 ^ + O) x3 + (2 i5 + Z)) x2 + (^ + 2 C) X + J5 + 2 D. 

.: 2 A + C = 0, 2 B + D = 1, A + 2 C = 0, B + 2 1) = 1. 

.•,A = C = 0, B = D=l/d. 

Substituting these values in (2) and integrating, we obtain (1). 

T, o r ^clx 1, (x2 + 1)1/2 1 

^^•2- I / I ix/ 9 I IX -o^og Ti — +:7tan-ix. 

J (x + 1) (x^ + 1) 2 "^ X + 1 2 

Here both Case I and Case III are involved ; hence, we assume 



(2) 



(X + 1) (X2 + 1) X + 1 X2 + 1 

Proceeding as above, we find A = — 1/2, B = C = 1/2. 
Substituting in (2) and integrating, we obtain (1). 

EXAMPLES. 
r* x^dx 1-x — 1,V2 ^ X 

/^ x^ dx r x'^dx 

J x^ + 3x2 + 2 ' J l-x^' 



f 



x^dx 1 , 1 , x2 — 2 X + 1 

+ 7 log 



X - 1)2 (x2 +1) 2 (X - 1) 4 ^ x2 + 1 

dx K (x + l)2 ^ 1 , 2.r-l 
— - log -^e ;- - -\ — ^ tan- 1 



x'^ + 1 "= x2 - X + 1 Vs Va 

/ dx _ 1 /^ dx 1 r* {x — 2) dx 
x3 + 1 ~ 3 J X + 1 3 J x2 - X + 1 ' 

/ (x - 2 ) (?x _ 1 /^ (2x-l)(?.r _ 1 /-» 3 dx 
x2 - X + 1 ~ 2 j .1-2 - .r + 1 2 I x2 - .r + 1 

/ 

/dx 1 , .c- -r x-r I . i , , _ ~ 



x^ + 1 (5 '''^ .r2 - .r + 1 ^/^ V^ 



1 , 2 X 



x -^ + X + 1 , 1 ,2x + l 

(X - 1)2 V 



(1) 



X _ A Bx -jr C ^ ,->. 



172 INTEGRAL CALCULUS. 



J x4 + 8x-^-9* ^' J ^* 



„ . dx ^ r a:2 + 1 ., 



/» (5 x2 - 1) dx 

J (x2 + 3) (x2 - 2 X + 5) 



, x2 — 2x + 5,5_^ ,x — 1 2^ ,x 



10 ' ^^ 



J (1+x 



)2 (1 + 2 X + 4 x2) 

_1 (1 + x)2 1^1 ^_. _^ 4x+l 

~3^°^H-2x + 4x2 3(l + x)"^3V3 V3 



,, /»x2 + 3x + l^ 4 ^ 2x-l 2 ^2x + l 

11. I —^—, r-— r- dx = --r: tail-l _ tan"! 7^ • 

J X4 + X2 + 1 V3 V3 V3 V3 

^^' J (x2 + a2) (x2 + 62) ^ a(62_a2) ^an-^ - + ^^2 _ 52) ^an-^ 



173. Case TV. To r equal quadratic factors of the denom- 
inator as (x^ +^932 + qy, there will correspond r partial frac- 
tions of the form 

Ax + B Cx + D _Lx -I- M 



(x^ +_/5a3 -h g')'" (cc'-^ H-^x + (j[Y~^ ^^ -\- 2^^ "I" ? 



(1) 



/•2 x3 + x2 -f- 3 X + 2 ^ 
Ex. j j^^^-^, dx 

= log(x2 + l) + |tan-ix + ^|^ 

Expressing the numerator in powers of (x^ + 1), we obtain 

2 x3 + x2 + 3 X + 2 = (x2 -fl) (2 X + 1) + (X + 1). 

2x3 + x2 + 2x + l 
x+ 1 

/2x3 + x2 + 3x + 2^ /'2X + 1 , , /" x-H ^ 



= log(x2 + l) + tan-ix-^^^^+J 



dx 



(X2 -j- 1)2 



(3) 



RATIONAL FRACTIONS. 173 



By example 23 of § 183 we have 

dx X , 1 



/ dx _ X 1 _j 

(a;'^ + l)2-2(x2 + l)'^2^^'' ""• ^^^ 

From (3) and (4) we obtain (1). 

From (2) we see that to resolve the given fraction into partial frac- 
tions by undetermined coefficients we should assume 

2x^ + x^ + ^x + 2 _ Ax + B Cx + D 

(X2 + 1)2 ~ (X2 + 1)2 "^ X2 + 1 

Note. The solution of the following examples should be deferred 
until after reading Chapter V. 

EXAMPLES. 
r 2xdx _1 a;2 + 1 1 a; — 1 

J (1 + X) (1 + X2)2 ~ 4 ^^ (X + 1)2 "•" 2 X2 + r 

r dx ^'^^ ^ ^^ \ 1 I 1 ' 

J X (X2 + 1)3 2 °^ 1 + X2 "^ 2 (1 + X2) "*" 4 (1 + X2)2 " 

„ /-xM-x - 1 1 - , o , ON I 2-x V2 ^ .X 

^- J (x2 + 2)2 ^^ = 2'''^^''^^'^^Ii^^^TJ)-^'^'' V^2' 

^ /'X5-2X + 1, , (x2+ 1)3/2 'Sx:^ + x + 2 3^ 

^- J X2(x2+l)2^^ = l^g^^ 2x(x2+l)-2^^^~'^- 

dx 



5 r____«i^_ 

• J (X-1)2(X2 + 



1)2 

1 1 ^i+ 1 1 1 ,1 

- loff — tan~ ' .r. 

4 ^ (X - 1)2 4 (x2 + 1) 4 (.1- - 1) ^ 4 



^ z' (4x2-8 x)(7x 3x2-x ,, (.r - 1)2 , 

6. I — ^' = \- loir ^^ + tan- ' v 

J (X - 1)2 (X2 + 1)2 (X - 1) (X2 + 1) ^ '^ =- 0-2 + 1 



CHAPTEE ly. 
INTEGRATION BY RATIONALIZATION. 

174. Rationalization by substitution. Chapters I and III 
provide for the integration of any rational algebraic differen- 
tial whether it is entire or fractional in form. 

In some irrational differentials we can substitute a new 
variable so related to the old that the new differential will be 
rational and therefore integrable. 

175. A differential containing no surd except a linear hasc^ 
as a + bx, affected ivith fractional exponents, can he rational- 
ized by assuming a + bx = tP-, ivhere n is the lowest common 
denominator of the several fractional exponents. 

For if a + hx = z^, x, dx, and the fractional powers of a -\- hx 
will each be rational in terms of z. 

Hence, the new function in z will be rational. 

Ex 1 C — 

■ ■ J U- 2)5/6 + (x- 2)^/3 

= 6 (X - 2)1/6 _ 6 log [{X - 2)1 '6 + 1]. 

Here the linear base is cc — 2, and n = Q; hence, we assume 
a: — 2 = ^6. 
.•.dx = Qz^dz. (X — 2)5/6 = 25^ (X — 2)2/3 = ^. 

dx /^ 6 z^dz _ r* zdz 

(x-2)5/6 -{-(x-2)2/3 - J z^ + z^~ J 2+1 

= Q[z-\og{z + l)-] 

= 6(x-2)i/6-6 1og[(x-2)i/6 + 1]. 



KATIONALIZATION BY SUBSTITUTION. 



175 



Here the linear base is x, and n = G ; hence, we assume 
X = z^. 

= f28-i29 = 1x4/3-^x3/2. 



j X Vo" 



+ 6x(lx 



EXAMPLES. 

2(2a-36x)(a + 6x)3/2 
15 62 



xcZx 



^ 2(2a-to) ^/- 



, = —77T7, Va + 6x. 



X 



(l + x)B/2 + (i+x)i/2 = 2tan-iVl + x-|. 
X (Zx _ 2 (2 g + 6x) 2 (2 + 6) Vg 



f^ (a + 6x)^V2 62Va + 6x &Vl + 6 

_ /* dx 1 . Va + 6x — Va , ^ . 

6. I — . = — F log ;= 1 when a > 0, 

J xVa + 6x Va Va + 6x + Va 

_ 2 ^ . /« + 6x . 

— , tan-i -V ' when a < 0. 

v=^ \ -« 

Z' xi/^dx 

J xi/2 + 2x--^/3 

= 7x2/3 _ Ig-i /i + ^^1 /3 - -^x} '^^^, loo- (1 + 2 xi z^'). 
4 2 b b 1() "^ 



Jo XI/- + XW.5 



(xi/^' + 1). 



/xi/o +1 (> 12 

-,,,.,, dx = - -— + -— , + 2 loo- X - 24 loo- (xi / '•- + 1^. 

xdx 

(2x + 2)«/4 + 4(2x+ 2)1/4 

= (2/5) (x + ;50) (2 x + 2) • / « - (4/;5) (2 x + 2) •>/••- 28 tan- '(^'--"^ ^- )' "'• 



176 INTEGRAL CALCULUS. 



176. A differential containing no surd except Vx^ -|- ax + b 
can he rationalized by assuming Vx^ + ax + b = z — x. 



For if 





^x' 


-h ax 


+ 6 = 


z 


-x, 








ax 


^b^ 


z' 


-2zx, 






^2 

^2l 


-b 

; + «' 


fjr^ 


2i 


{z" + az 


+ b) 








{2z + 


af '^^' 










■ z 


z 


'' + az-\-b 
2z + a 




^x" 


+ ax 


-\-b = 



Hence, x, dx, and Vic^ -\- ax -\- b are each expressed ration- 
ally in terms of z. 

Ex. I — , = log (1) 

J ic Vx-^ + a; + 1 Vx2 + X + 1 + x + 1 

Assume Vx^ + x + l=2; — x; - (2) 

then X + 1 = 22 — 2 X2:, 

_ g2_i _ 2 (z2 +' g + 1) dz 

^~22 + l' '^''~ (22 + 1)2 

Vx2 + X + 1 = 2 - X = ^\"^ ^, "t ^ • 
22+1 

/ ( Z X _ r__dz__ _ 2 — 1 

XVX2 + X+1" J 22-l-^°S^ + l- 

Substituting for 2 its value in (2), we obtain (1). 



177. A differ eritial containing no surd excejjt v — x^ + ax + b 
can be rationalized by assuming 



V- x^ + ax + b = (/? - x) z, 
where ^ — x is one of the factors of — x^ + ax + b. 

Denoting the other factor of — x^ -\- ax -\- b hj x -^ y, assume 
■\/-x^-\-ax-^b = V(/3 -x)(x-h y) = (^~x)z; (1) 

P 2 

then x -\- y = (13 — x) z^, x= ^,7 ' 

Hence, x and therefore dx and V— x'^ -\-ax + b are expressed 
rationally in terms of z. 

Note. In § 176 the coefficient of x2 is + 1 ; in § 177 it is — 1. 



EXAMPLES. 

1. C ^^ == = 2 tan- 1 (X + Vx--^ + 2x-l). 
J xVx^ + 2x — 1 

^ /• dx V2 , Vx^ — X + 2 + X— V2 

2. I , = ^7- log , ^ • 

J X Vx2 -x + 2 ^ Vx2-x + 2 + x+V2 

J— ^ (^X =: log (X + 1 + Vx2 + 2 X) 



X + Vx2 + 2 X 



J (2 + 3 X 



dx V2 , V4 + 2 X — V2 — X 

=^=: ^^^ — ^ — log — — r^=^ 

) V4 — x2 8 V4 + 2x + V2 — X 



Assume V4 — x^ = V(2 — x) (2 + x) = (2 — x) z ; 
then 2 + X = (2 — x) ^2, 







X 


2^2-2 
~ 22 + 1 ' 


dx = 

4z 

~ ^2 + 1 ' 


Szdz 

(^2 + 1)2 


' 




V4- 


-X2 


= (2 - X) 2; = 








2 + 3x 


822-4 

£2+1 










f?X 




_1 r C?2 


V2 

1 8 




-1 


(2 + 3 


x)V4^ 


-X2 


~2j 2z^- 


+ 1 



V2 , V4 + 2 X — V2 - X 



8 V 4 + 2 X + V 2 - X 

f^ dj '" 

xV2^ 



^ , ...^ V2 , V2 + 2 X — V2 — X 
5. I — , = -r- If^g , ,. . 

^ V2 + 2X + V2-X 



X — x-^ 



r-"' (?x ^ _ 2/2-.ry /2 2 V 

Jo ( 1 + X) V2 + X - x2 ~ 3 V 1 + X ) 3 



8 r ^ ^^^ =..J .r + Vx2 + x + l . .3 + vgY 

J, (l+x)Vx2 + x+l "^ V2+x+V.r2 + x + l l+%3/ 

/••'• X (fa 3 + x V3 



2x-x2)8/a 4V8 + 2X-X'-* 4 



Assume V ;> + 2 X — x- = (3 — x) z. 



178 INTEGRAL CALCULUS. 

178. Irrational differentials of the form x™ (a + bx")''/^dx, 
ivliere r and s are integers and s is 'positive, can he rationalized 
by assuming, 

T .1 7 m + 1 . 

1. a + bx^ = z^, wfien is an integer or zero. 

TT 11 7 m -f 1 r . 

11. a + DX° = z^x^y wlien ■ 1 — is an integer or zero. 

Assume, a + hx"^ = z^ ; 

then {a^hx'y = z% CI) 



( 



z" — a 



(2) 



dx^-z^-\---\ dz. (3) 



s J z" — a^ 



hn \ h 
Multiplying (1), (2), and (3) together, we obtain 

ni + 1 



s , , f z'' — a" ^ 



x'^ {a -f- hx'^y/'dx = —z'' + '-^ [ — ^^ J dz. (4) 

The second member of (4) is rational, and therefore in- 
tegrable, when (m + 1) /n is an integer or zero ; hence I. 

Assume a -\- hx"^ = z'x"", or cc" = a (z' — h)"'^ ; 

then (a + bx'') '' / ^ = (z'x'') '• / ^ = z^'a'' ' ' (z' — b)-''^ % (1) 

ic = ai/«(^« — Z»)-i/'*, a:'" = a"'/ «(;s^ — ^>) -"*/", (2) 

dx = --a'/^'z'-^ ('-'« - by-~^dz. (3) 

Multiplying (1), (2), and (3) together, we obtain 

r ^ ffl + 1 r / w + l r \ 

cc"* (a + bx^'fdx = « '^ ^{z' — b) V « « ; ^'• + ^-1^?^. 



The last member is rational, and therefore integrable, when 
— h - is an integer or zero ; hence II. 



BINOMIAL DIFFERENTIALS. 179 

EXAMPLES. 

1. Cx^a + 6x3)-i/2^x = ^(a + 6x3)3/2 _ 1^ Va + 6x3. 

Here = — ;; — = 2, and s = 2 : hence, we assume 

w 3 ' 

a + 6X3 = ^2. . ((^_^^^3)-l/2 = 2-l, (1) 

0:6= Af_^^. ... 6x5(^x = -(22-a)zd2. (2) 

Multiplying (1) by (2) and dividing by 6, we obtain 
x^dx 2 r, . . , 2 „ 2 a 



/ x^c^x _ 2 Z', 2_ \w -_?_ 3_ 
(a + 6x3)1/2 -3 62 J ^^ «)^^-952^ 



3 62^ 



= ^(« + ^^')'''-|^(«^+^^')^'''- 



^ f dx 1 , X 

2. I — , = - log 

J xVa2 + x2 «• Va2 + x2 



+ a 



_.-. m + 1 — 1 + 1 ^ . ^. 

Here = = 0, and s = 2 • hence, we assume 

n 2 

a2 + x2 = 22 . 

.-. xdx = z dz, X- = z^ — a2. 
dx r dz 



/ dx _ r dz_ J_ z — a 
xy/oC^ + x2 J 2" — a2 ~ 2 a ^2 + a 



1 , Va2 + x2-a 

T— log , ■ 

^ « Va2 + x2 + a 

1 , X 

I02: 



« " Va2 + ^2 + a 

3. I — - log ; = 

J xVa2 — x2 i^ va2 _ 3.2 + ^ 

r-'- .r'v/.r _ 4-3 x^ £_ ^ 

• X (^-'>'t^')''-~0V2-3x2 1)V2' 

r-'^ X8C?X 

Jo V(?, + 6x-2 



X^cto 6.1-2 _ o (J, 



., ,,. va + 6x2 + - .,,.,- 
,> 6- 3 6^^ 



180 INTEGRAL CALCULUS. 



J X 



dx (2x2-l)Vl +x2 

Wl + X2 ~ 3X3 



-_-. m + l.r — 4 + 1 1 „ , n , 

Here 1- - = = — 2, and 8 = 2; hence, we 

?i s 2 2 

assume 

.-. X = (22-l)-l/2, 3.-4 = (^2 _ 1)2^ (1) 

dx = -{z^-l)-^/-2zdz, (2) 

(l+x2)-l/2=^-lX-l = Z-l(22- 1)1/2, (3) 

Multiplying (1), (2), and (3) together, we obtain 



/^,(i+'^.)i,. = -J(^^-i)<'^ = ^-f' 





where 




Z=Vl+x2/X. 


7. 


1 (1 


acZx 


ax 


+ x2)3/2 


Vl + X2 


8. 


Jo a 


dx 


. x(2x2 + 3) 


+ x2)5/2 


3(l + x2)3/2 


9. 


Jo (^ 


X2dx 


X3 


; + 6x2)5/2 


3 a (a + 6x2)3/2 


in 


r 


dx 


a + 2 6x2 




J ^^' 


(a + 6x2)3/2 a2x (a + 6x2)1/2 


11 


r 


(^X 


3 x3 + 2 a 



+ x3)2/3 



CHAPTER V. 

INTEGRATION BY PARTS. REDUCTION FORMULAS. 

179. Integration by parts. By differentiation we have 

d (uv) = udv + vdu. 
Integrating both members and transposing, we obtain 

I udv = uv — i vdu. (1) 

The use of formula (1) is called integration hy parts. 

In applying (1) to particular examples the factors u and dv 
should be so chosen that dv is directly integrable and vdu is 
a known form or one easier to integrate than udv. 



I X log X ( 



Ex. 1 X log xdx = ^-r log a: — — 



Let u — log.x ; then dv = x dx, 

dx. = dx/x, V = X-/2. 

Substituting in (1), we obtain 

<ix 

X 

(.rV'2)loo-.f-.r-/-l. 



j log X ■ X dx = \og X • 7j — J \^ 



(2) 



In (1) the second product can be obtained from the first by 
integrating its second factor, and the third ]n'oduct from tlie 
second by differentiating its iirst factor. 

The following examples will illustrate how we can bv the 
use of this law abbreviate the applications of \^\). 



182 INTEGRAL CALCULUS. 



Jx.cos... = ..si„.-J, 



Ex. 1. i X ■ cosxdx = X- sinx — I sinxdx 
= X sin X + cos X. 



X • e«^cZx = x i dx 

a J a 

Ex. .3. Txs {a - x2)i /2 (ix = Cx^ ■ {a - x2)i / 

1 
3 

g ,j - ^ , 15 



■xdx 



= x2 [-J(a-x2)3/2] + r|(a-x2)3/2.2xdx 

= - — («- X2)3 / 2 _ A ((J _ 3-2)5 / 2. 



EXAMPLES. 

1. I log xdx = X (log X — 1). 

2. Jx»logxdx = f^(logx-^); 

x** log xdx = — - — • § 86, example 6 

v^ + ^) 

3. J X sin xdx = I — x cos x + sin x I = 7t. 

4. I sin— 1 xdx = I X sin— ix + Vl — x2 I = — — 1. 

5. I tan-ixdx = xtan-ix — -log(l + x2) = — — 

6. Jcos->x.x = xcos-.x-Vrr^. 

7. j cot-ixdx = xcot— ix 4- ^log (1 + x2). 



log 2 

2 



INTEGRATION BY TARTS. 183 

//— -, xVa^ — x-2 , a2 . X 
Va^ — x^dx = h — siu- 1 - • (1) 

^ Ji (I 



/Va^ — x^ . dx = Va'" — x2 . X 4- j /' ^ '^ 
■ J Va2 - : 



rt2 — (a2 — a:2) , 



10, 



= X Va2 _ ^2 4- a2 sin— i I Va2 — ^2 cZx. 

«^ J 

Transposing the last term and dividing by 2, we obtain the result 
in (1). 

9. r Vx2 -a^dx = ^ Vx2 - a2 - ^ log (x + Vx2 - a^). 

C Vx2 + a2 dx = ^ Vx2 + a2 + ^ log (X + Vx2 + a2). 

11. I x2 sin xdx = 2 x sin x — (x2 — 2) cos x. (1) 

I x2 • sin xdx = x- • (— cos x) + 2 j cos x -xdx (2) 

I X • cos X (Zx = X • sin x — j sin xtZx 

= X shix + cosx. (3) 

Substituting in (2) the value in (3), we obtain (1). 

13. I x2 cos X (Zx = 2 X cos X 4- (x2 — 2) siu x. 

14. I X'" cosxdc = X"* sin.r — //I j x'"-' siiiX(Z.r. 

15. I vcrsin— 'x(?x = (x — \^ vorsin-ix + ^''1 x — x"-. 

X(;'''(Zx = — (ex — 1). 



184 INTEGRAL CALCULUS. 

17. Cx^e'^-dx = y (^' - ^ + I) • 

18. I x'"ec^dx = I x'^^-^ e"^ dx. 

19. I x^e^^^ dx = — I x3 \ — ) • 

J c \ c c2 c3 / 

or. r ^ ^ a"" r , 3 • x2 , 3 • 2 • X 3 • 2 ■ l-I 

20. j x^a^dx = -^ x3 - + 7^ -, - • 

J log a L log a (log ay (log afA 

21. I = H I dx. 

J x'» m — lx"'-^ m — lj x'"-i 

_„ /» , -, e^^logx 1 /»e«'3= ^ 

22. I e^-^ log X c?x = ^ I — dx. 

J c cj X 

/j-w + i r 9 2 n 

24. I X COS— ixdx= - cos— ix — - Vl — x"^ + -sin— ^x. 
J 2 4 4 

25. r x2 sin- 1 X dx = ^ sin- 1 x + - (x^ + 2) Vl — x^. 

/X^dx /I \ y 

— — — - tan— 1 X = f X — - tan— 1 X j tan- 1 x — log Vl + x^. 



180. Additional standard formulas. For convenience of 
reference we write below the formulas obtained from examples 
8, 9, and 10 of § 179, and examples 2 and 3 of § 178. 

r Va2 _ 1,2 ^^ = !f V(^2 _ ^2 _p ^ sin-i - 4- a (1) 



du 



u /— : a 



= - ■\/ii^ ± a' ± - log {u + V^2 ^ ^2^J _^ ^_ (2) 
/-%=4log--^==+C. (3) 



REDUCTION FORMULAS. 185 

181. Reduction formulas. A formula by wliich any inte- 
gral not directly obtainable is made to depend upon a standard 
form or on a form easier to integrate than the original func- 
tion is called a reduction formula. Thus, the formula for 
integration by parts is a general reduction formula. 

Many special formulas are obtained by applying this general 
formula to particular forms. 

182. Reduction formulas for j x"^ (a + bx^)p dx. 

C x"^ {a + hx^'Y dx 

=—rr-\ — 1^^^ rh~L — ^ri^ x^'^-\a^-hxy^dx, 

(A) 

x"' + '^ (a -\- hx"")}' anp r , . ^ , . 

or ^, , / + , ^ , , x^{a + hxy^-^dx, (B) 

' j .r"' + "(r^^-/>.r")^'(7.r 
or \ / + ^ . ... y\a + />.r") " + > dx. 

Formula (A) decreases m hj 7i. 
Formula (B) decreases ^> b}' 1. 
Formula (C) increases n) by u. 
Formula (D) increases p by 1. 
Formulas (A) and (B) Tail wIumi up -f /// + 1 = 0. 
Formula (C) fails wlion m -\- \ - 0. 
Formula (O) fails wIumi /> H- 1 0. 

When (A), (r»\ or (C) fails, Ww motlunl c^f § 1 7S is appli- 
cable. AVhen (D) fails, /> =^ — i niul provious mot hods api>ly. 



xJ"' + ^(a + hx:"y + ^ h(np-{-m + n-{-l) 
a (m -\- 1) a (in + 1) 

(C) 



186 INTEGKAL CALCULUS. 

183. Proof of formulas (A), (B), (C), and (D). 
Integrating by parts when w = a?"*"""^^, we have 

= A — —^ TT — 7-iT I x'"-''(a-{-bx'')P-^^dx. (1) 

nb(p-\-l) nb(p-\-l)J ^ ^ ^^ 

Cx""-"" (a + bx'')P + '^dx = Tie'"-" (a + &x")^ (a + bx"") dx 

= a Cx^-"" (a + bx'^ydx + b Cx"' (a + bx")Pdx. (2) 

Substituting the last member of (2) for its equal in (1), 
and solving for j x"" (a + bx'')Pdx, we obtain (A). 

Solving (A) for i x^~'' (a -\- bx^)^ dx, and substituting in 
the resulting identity m -\- n for m, we obtain (C). 

To;"* {a + &:c") ^ ^x = Tic^" (ci + ^x") (a + bx^) ^-^dx 

= a Tx'" {a + bx'^y-^dx + 5 rx'» + '^ (a + bx^'y-^dx. (3) 

Substituting, in (A), m + tz for m and ^ — 1 for ^, we obtain 

a;^ + Vc^ + ^^")^ a(m-hl) C / , 7 X , 7 /.x 

= , , \_ — fr- - , , ^, _/ix he" (^ + ^'x'^)^-^c?^. (4) 

6 (np + m + V) b (np + m -\-l)J ^ ^ ^ ^ 

Substituting in (3), and combining similar terms, we 
obtain (B). 

Solving (B) for j x"^ (a -\- bx'^y-^dx, and substituting p-\-\ 

ioT p, we obtain (D). 



REDUCTION FORMULAS. 187 



EXAMPLES. 






xHx X /— , a?- . ,x 

^2 2 2 a 



Here m = 2, n =^ 2, p = — 1 /2, a = a'^, b = — 1. 
Substituting these values in (A), we obtain 



x^{a^ — x^)-^/^dx = — ^ I —-=z 

-2 —2J Va^ 



x^ 



2 2 a 

We use formula (A), because decreasing m by n in the given dif- 
ferential reduces it to a know^n form. 



Va2-x2 \4 ^2) 



x^dx {x^ , Sa^x\ /—. -c , 3a4 . .x 

4-2 a 



C x'^dx X /-r— ; — - a2 ^__ — _ 

3. I , = - V x2 ±(jfizii— log (x + Vx- ± a^). 

J Vx2 ± a2 2 -^ 2 ^^ 



/ (^x _ _ vg^ — x2 _1_ ^ X 

x3Va2-x2 " i^ (t'-c' 2 a3 ^S v^, _ ^^, ^ ^ 

Apply (C), and then use (3) of § 180. 



r dx _ vx2 4- aP- 1 X 

J x3Vx2 + a2 ~ 2 a%2 2 a^ ^^ a + Vx^ + a^ 



/f?x Vu:- — a- ,1 , X 
/ = ,, ., ■> f- :r— I sec- 1 - 

dx V X- ± a'- 



r dx _ 
X- Vx- ± d^ 



a-x 



r^ dx _ _ Vg- — X- ^ 
^„ x'-^Va^ — x" '"^"-c 

9. j x" V(i- — X- dx = '^ (2 .r- — a-) V(i- — .r- + -V : 
Apply (A), and then use (1) of § 180. 



188 INTEGRAL CALCULUS. 

10. f.V^u. 



= f (2 a:2 ± a2) Vx^ ± a^ - ~ log (x + Vx^ ± a^). 



V x'^ — a^ 

X 

Apply (B). 



11. i —dx = Vx'2 — a^ — a sec-^ 

J X a 



12. f "*" ^' dx ^ Vx^ + a2 + a log f 

c/ ^ a+ Vx2 + a2 

J^^Va^ — x2 -T, 
(ZX = V a2 _ ^2 -[- (;j iQg 
a ^ 



a + V a2 — x2 
14. r(a2 - x2)3 /2 dx = ^ (5 a2 - 2 x2) Va2 - x2 + 



3a4 . .,x 
o a 



^^ C Va2 _ a:2 Va2 - x2 . r 

15. I 1 (^x = sm— 1 - 

J x^ X a 

Apply (C), and then use (1) of § 180. 



/Vx2 ± a2 Vx2 ± a2 , ^ ^ , / , , ^ , 
^ dx— 1- log (x + Vx2 ± a2). 
X X 



16 

/^ t?X _ Jg 

• J (a2-x2)3/2 ^2V^^Z:^ 

Apply (D). 



/^ rtx _ ±x ^ 

• J (x2±a2)3/2 ^2Vx2±a2 

n xHx _ X . _jX 

!«• J (a2-x2)3/2-v^^zr^, «^^ a' 

J (x2±a2)3/2 Vx2±a2 
21. r(x2±a2)3/2^x 



.2 -U ^2 _L § , 



- (2 x2 ± 5 a2) Vx2 ± a2 + - a* log (x + Vx2±a2). 
o o 



... f 



REDUCTION FORMULAS. 189 

dx _ X (3 a2 + 2 x^) 

{X^ + a2)5 /2 - 3 ct4 (3^2 + a2)3 /2 ' 



J (^^ + «^)^ ~ 2 a2 (x2 + a2) "^ 2 a3 ^^ a ' 



(x2 + a2)3 ~ 4 a2 (x2 + a2)2 + 8 a* (a2 + x2) "^ 8 a^ ^^^ a 



2^- • -^.,. 



/ ax 
(x2 + a 

_ 1 X 2 >i — 8 /^ dx 

~ 2{n — 1) a2 (x2 + a2)7i-i 2 (n — 1) a^ J (^2 + cC^yi 



f 



X — a /- ; , a^ . ,x — a 



26. i V2 ax — x2 dx = — - — V'2 ax — x2 + 



2 2 a 

Reduce the expression to the form in (1), § 180. 

27. I x'»V2ax — x2dx = \ x"* + 1 /2 V2 a — x dx 



( x'»V2ax — x2dx = j x"* + 1 /2 V2 a — X 

■I 



x'«-i (2 ax — x-Y '" , (2 m + 1) a r , /"T ; , 

= ^ — rr^ — — + ^ r^ i .^'"-^^2 ax — x- Jx. 

?n + 2 m + 2 



, x'^dx _ x»'-W2ax — x2 , (2///— l)a (^ x">-'^dx 



29. 



/ x'^dx _ _ x"'-iV2ax — x2 (2/// — l)a /* 

V2 ax - X- "^ ''' J 

/> xd^_. 30. f-^ 

J V 2 ax — x2 J ^/2 c 

/ dx 
x"»V2 ax — x'- 



V2 ax — x'- ?>) — !_ /• (?x 

(2m - l)ax"' (2 m - l)a J r " - 1 V2 ax - x'-^ 



CHAPTER VI. 
INTEGRATION OF TRIGONOMETRIC FORMS. 

184. j tan^uduor j cof^udu, n any integer. 

When n is a positive integer 

I tan" udu= | tan""^ u (sec^tc — 1) du 

I idun!^~'^ u du. 



tan'' 



By repeating this process the integration of i^n^udu is 
made ultimately to depend upon the integration of tan u du or 
du according as n is odd or even. 

When ^ is a negative integer, as — m, we have 

tan~ "* u du = cot'" ti du, 

which may be integrated by a process similar to that used 
above. 

Ex. I taii-4|dx= I cot4^cZx= I cot2-f csc2-— Ijdx 

= -S fcots^dcot^- Ccot^^dx 

= -C0t3^- r(cSc2|-l)cZx 

= — cot3 (x/3) + 3 cot (x/S) + X. 



TRIGONOMETRIC FORMS. 191 

185. I sec'^uduor j csc"udu, n even and positive. 

I ^QQ^udu= I sec^'^^i^. sec^ udu 

= C(tSin^u + iy^-'^'^-dtiimi, (1) 

which can be expanded and integrated directly when 7i is even 
and positive. 

Ex.Jsec«...=J(ta,.. + l,..tanx 

= tan^x/S + 2 tan^x/o + tanx. 

186. j tan"^ u sec" u du or j cot"^ u csc" u du, m odd and 
positive or n even and positive. 

j tan'" w sec" itdu = j (sec^^^ — ly^'^^^^^sec""^ w • dsecii, (1) 
or Ttan'" it (tan^ u + 1)(« --') /2 . ^ tan u. (2) 

The form in (1) can be expanded and integrated when m 
is odd and positive ; tlie form in (2) can be when 7i is even 
and positive. Compare (2) with (1) of § 185. 

Ex. 1. I tan^ X sec^ X (Zx = | (sec^ x — 1) sec'* x • (^ sec x 

= sec" x/7 — sec^ x/T). 
Ex. 2. j COt5/2a;csc4 3;t^^j^ = — i cot5/2.c(cot-x + 1) • (7cotx 

2 *^ 

= — 77 cot ^1 /- X — ^ cot " /- X. 
11 / 

EXAMPLES. 



1 A .? J tan-x , , 

1. I tan'\X(iJx = — h log oosx 

2. I (air xax = — ._ tan .)• + . 



) 




INTEGRAL CALCULUS. 




3. 


1 tan^ xdx = 


tan4 X 


tan'^x , , 




4 


2 ~r log secx. 




4. 


Ctsin^xdx- 


tan^ x ■ 
5 


tan^x 
- .3 + tan X x. 




5. 


Csec^'Zxdx- 


tan- 2 ; 
14 


z 3 tanS 2 x tan^ 2 x 
"^10 ' 2 ^ 


tan2x 

2 


6. 


i csc6 ^dx = 


= _|eot^|-|cots|-2oot|. 





7. JtanSxsec-x.. = -sec-e./3 + sec-./5 

= cos^x/S — cos^x/S. 

/3 6 3 

tan^x sec5/3x(Zx = — - seci'^/^^ _ _— gecH/Sx + - bqq^^^x. 
17 11 5 

9. Aan' /2 a: sec* a; (Zx = ^ tani^ /2 ^ + | tan^ /2 x. 

^^ /'sec^xdx ^ ^ ^ cot^x 

10. I — — - — = tan X — 2 cot x — 

J tan^x 3 

r .r, A -, cot^x cot^x 

11. I cot5xcsc*xdx = — • 

14. j (sec X + tan x)* dx = - (sec^ x + tan^ x) — 4 sec x + x. 

15. j (tan X + cot x)-^ dx = - (tan^ x — cot^ x) + 2 log tan x. 

187. j sin°^u cos^udu, m or n odd and positive. 

I sin"' u cos" udu= j sin"* i^ (1 — sin^2^)~2~ cZ sin u, (1) 
CDS'" ?i (1 — cos^ zi) 2 dcosu. (2) 



TRIGONOMETRIC FORMS. 193 

The form in (1) can be expanded and integrated when n is 
odd and positive; the form in (2) can be when m is odd and 
positive. 

EX.1. Jsine«.cos.3x.x=Jsine.s,(i_3in.,),,i„x 



5 5 

- sin^ /^ X — — sinis /s x. 

o lo 



' cos^xda; _ /^ (l — sin'^x)d smx _ ^^ 1 

sin* X sin X 3 sin^ x 



Ex.2. r^t^= f< 



COS"* + "wc?i^ 
sec-^'" + '*h«c/w, 



188. j sin™ u cos'' u du, m + n even and negative. 

/r sin"* it 
sm'" i^ cos" i^ (XM = I 
J cos"* It 

= r tan'" it 

which is directly integrable by the method of § 186 when 
m + i^ is even and negative. 

/C OS" X ^ 

' . \. dx = i cot^xcsc^xcZx 
sm^ X J 

= — j cof^x(cot-x + l)d cotx 

COt^ X COt^ X 



Ex.2, j cos-3/6a:sin-' '(^xdr = ^ cot-'^/^x • csc- 



r dx 







— " 


- - cot- / 


'^x. 


' dx 
sin-^x 


= — cot X — 


• (t:\n-.( 

C'Ot.« X 


• + \)d t 


an X 



194 INTEGRAL CALCULUS. 



EXAMPLES. 

1 2 

1. I sin^ xdx = —-cos^x + -cos^x — cosx. 



2. i sin^xcos^xcZx = -sin^x — -sin^a;. 



3. Jsin^xcose.,.. 5. Jcos-.sin3.<to. 

4. Jcos^xsiae..^. 6. JcosS . si.- x dx. 

7. I — r-dx= I tan^xsec^xdx = — 1 — 

I cos^x J 5 3 

8. I sm^x Vcosxdx = — 3 Vcosx( ^^ 1 7^)' 

9. r ^ 

/3 3 

tan— ' /-^xsec^xdx = -td^n^^^x — Tcot^^'^x. 
2 4 

^ 

sin^ X 

/ sin3/2x(Zx _ 2tan5/2x /'cos^^^ 

COS' ''2 a; 5 ' • ^ sin^/' 



10. I -T-7, r~ = — ;; H 2 tan x — cot x 

sm^ X cos* X 3 



^ ^ , dx tan* X , 3 tan2 x cot^ x , „ , 

11. I -7-; ^ = — \ 1- 3 log tan X. 

cos^x 4 2 2° 



189. j sin°^ u cos^ u du, by multiple angles. When m 

and n are positive integers, sin"* u cos" u du can be expressed, 
by means of trigonometric formulas, in a series of the first 
degree in the sines and cosines of multiples of \i. 

Each term of any such series can be integrated directly. 



TRIGONOMETRIC FORMS. 195 

Ex. 1. j cos^xcZx = ^ I {I + G0s2 x) dx = - + ' — j 

Ex. 2. Ccos^xdx = - (*(! + cos2x)^dx 

= ^ C[l + 2cos2x4-|(l + cos4x)]dx 
= 3x/8 + sin2x/4 + sm4x/32. 

We shall use this method only when that of § 187 fails ; 
that is, when m and n are both even. 

In any such case the trigonometric formulas for shi^ u, 
cos^ u, and sin u cos u in terms of sin 2 ic and cos 2 ic will 
enable us to transform all terms with even exponents so that 
they can be integrated by previous methods. 



1. j sin^xdx 



EXAMPLES. 

X _ sin 2 X 
2 4 

1 / 3 X . ^ , sin 4 X " 



/ 

/• A 1 1 / 3 X . ^ , sin 4 X \ 
sm'^xdx = 7 { ":^ sill 2 X H — j 

/. P , 1 /- ^ • n , sin3 2x , 3sin4x\ 
sin^ X tZx = — ( 5 X — 4 sin 2 X H :, 1 j 



4. I sni^ X cos* X CZX = — ( ;;, + X 7 — j • 

sin"-^ X CDS'* X = (sin x cos x)- cos- x = sin- 2 x (1 + cos 2 x)/S 
= sin- 2 X ■ cos 2 x / 8 + (1 — cos 4 x) / l(!. 

_ r* . , ., , 1 / siii'^2x , sin4x\ 

5. I sin-*xcos-X(Zx = -- ( . h x z — ) • 

J l() V ;. 4 / 

6. I sin-xcos-X(tx = "(•>' , - ) * 

7. Jsin^rcos»X(/.r = ^^(^^ - sin 4 x + '^'^J^-'') • 



196 INTEGRAL CALCULUS. 

190. Reduction formulas for j sin"i x cos'^ x dx. 



siTL^x Gos^ xdx 

_ sin^-^cc cos" + ^a3 , m — 1 
7)1 + n 

sin"' + ^ cc cos" ~ ^ cc , 71 — 1 
or , h 



7?^ — 1 /• . „ 

H ■ — I si-nJ^~ - X cos"" X ax, (1) 

771 -j- nJ ' \ y 

j sin'^x cos''~^xdx, (2) 

m + 71 VI + 71 J ' \ / 

/cos'^xdx _ cos" + ^a:^ m — ?i — 2 rcos^xdx 

sin"'a3 (7/»i — 1) sin'"~^a; ??^ — 1 J siii"'~^cc ^ ^ 

/sin'"a:'<:Z£c_ sin"' + ^£c ti — 7n — 2 rsm"'xdx 

COS"X (/i — 1) C0S"""^iC 71 — 1 J C0S"~^a3 ^ "^ 

191. Proof of formulas in § 190. 

Letting u = sin'"~^cc, and integrating by parts, we have 
j sin"^ X cos^ X dx 



— j sin"'-^^ (3QsK + 2^^^_ 
sin'"""^cc cos" + -a? = sin™~^£c cos 



71 -{- 1 71 -\- 

sin'"""^cc cos" + -a? = sin™~^£c cos^cc (1 — sin^cc) ; 



g^j-^m-2^ gQg« + 2 ^^^ 

= I sin'"""^ a; cos^ccc^a? — | sin'^cc cos"£CfZa:. 

Substituting this value in the first identity, and solving 

for j sin'^cc cos" xdx, we obtain (1). 

Letting u = cos'^^^cc, and proceeding in a similar manner^ 
we obtain (2). 

Substituting 2 — m for 771 in (1), and solving for I — ^— — ? 

we obtain (3). 

^ — ' 

cos X 

we obtain (4). 



TRIGONOMETRIC FORMS. V.)\ 

EXAMPLES. 

/• . , sin'" — ixcosx , m — 1 /• . 

1. I sm'«xdx= 1 I sin'«-2x(Zx. 

J m m J 

Putting for n in (1) of § 190, we obtain the formula above. 

2. By the formula in example 1 show that 

/• . , ^ sin3 X cos X , 3 , 

(a) I sni* xdx= + ^ (x — sin x cos x). 

,-, r ■ f 7 cosx /sin^x , 5 . „ , 5 . \ , 5x 

(b) I sm^ xdx= — ( — 3 \- — sm-^ x + - sm x j + — - • 

/* , sinxcos"— ix , n — 1 /* „ , 

3. I cos'^xcZx^: 1 I cos"— 2xdx. 

J n n J 

. r , , sinx/ „ ,3 \ , 3x 

4. I cos* xdx = -—- 1 cos'^ ^ "^ 9 '^'^''^ ^ ) "^ "«~ ' 

/^ (ix cosx m — 2 n dx ^ 

J sin"«x (m — l)sin'"— ix m — lj sin'« — 2;^: 

/• r •, COS X / 1 , 3 \ , 3 , ^ X 

6. I csc^ xdx = ( -^,~ 4- ^-, . ^ ) 4- - log tan - • 

J 4 Vsm-^x 2 sni^x/ 8 ^\ 2 

/* fte sin X n — 2 /» 

J cos"x (n — l)cos"-ix ?i— 1^ ( 



— 1 I cos«-^x 



8. |sec'X(^x=- :r'(7J 1 ^77^ T" + o ) 

J 2cos^x\3 cos'^x 12cos2x 8/ 

+ (5/ IG) log (sec X + tan x). 

/» . „ . , sin X / cos^ X , cos^ x , cos x \ , x 

9. j sm2xcos4xdx= ^- (- - . - + ^^ + -^ ) + ^- 

_ _ /•sin* X , sinS x , , „ 3 . 

10. I — ^ dx = f- sni" X cos X — - (x — sni x cos x) 

J cos-x cosx 2 

sin'^x 3 , 

= :(x — sm X cos .r). 

cos X 2 ^ ' 

/»cos'*x , cosx ,., ., , 3x 

11. I -^^7— dx = — -' . {'> — cos- X) — — - • 
J sin- X 2 sni x 2 

,^ /* dx 1 cosx , 3 , X 

12. I r— — = , ■ .. + - lou- tan - • 

J sni'«xcos-x cosx 2sin-x 2 ^ 2 



198 INTEGRAL CALCULUS 

du 

a + b sin u 



192. f l^ and f 

^ a + b COS u J i 

a -\-h cos u = ai sin^ - + cos^ o ) "^ ^ ( ^^^^ o ~ ^^^-^^ 9 ) 
= (a-b) sin^ (zi/2) + (a + ^>) cos^ (ii/2). 



/ du 2 r Wa — bdt3.T\(u/2) 
a -\- b cos u Va — bJ i 

2 , f \a — b , It 



a -\-b cos ?^ Va — bJ (cl — b) tan^ (^/^) + («^ + ^) 



(Vj 



^ tan-i \ — — tan - ; (1) 

Va^ - b^ \^a + b 2y' ^ ^ 



or 



(/?/ _ -2 ^ -^h-adt2Jii(it/2) 



a -h b cos z^ ->/^ _ f^J (b — a 



(b - a) taii^ (u/2) - (b -h a) 



1 , VZ> — a tan (w / 2) + V& + a ^^^ 

log ■ ^ ^ ^ 7=- (2) 

^b^-a^ -\/b-atSin(u/2)-^b-\-a 



a + 6 sm z^ = a ( sm^ - + cos'' - j -\- 2 b sm - cos - ■ 



i) 



r du _ r sec^(?^/2)r7^^ 

' * J a-{-b sin u J a tan^ 0^/2) + 26 tan (u/2) + «- 

g seG^u/2)du/2 
t3.n (u/2) ■+■ bf -\- (a' - b^) 



'J [^ 



2 . a tan(i^/2) + 6 ._. 
tan-^ ^ ' J (3) 



-s/a? - b^ -J a? - W 



1 61 tan fa/2) + 5- V^^-^^ 

'''" V^^ZT^ ""^ a tan (u/2) + b -h V^^^Z^' ^ ^ 

If a > 6 arithmetically, we use the forms in (1) and (3) ; 
\i a<Cb arithmetically, we use the forms in (2) and (4) ; that 
is, in each case we use that form of the integral which is real. 



TRIGONOMETRIC FORMS. 199 

EXAMPLES. 

/ dx 1 r d{2x) 1 ^ , .^ ^ ^ 

5-3COS2X = 2 J 5-3COS2X =A ^^^"' ^^ ^^^^ ^^^ 



f' 



„ . dx 1 ^ - 5 tan x + 4 

2- I ;; I . ■ o = o tan- 1 

5 + 4 sm 2 X 3 3 



o f ^^ ^1 , tan(3x/2) + 2 

J 3+5cos3x 12 ^tan(3x/2)-2 

/* dx _ 1 2 tan(8x/2)+ 1 ^ 

J 4 + 5sin3x~9 ^^2 tan(3x/2) + 4 ' 

dx 1,2 tan x + 1 



/^ 



= QlOg 



5 cos 2 X 8 2 tan x — 1 



193. Integration of trigonometric forms by substitution. 

Assume sin x = z; then 

cos x = (l- zy\ dx = {\- zY^'^dz. 

.'. Tsin"^ X cos" X ^x = Cz'"" (1 — r^Y"-^^'^dz. 

The last form is integrable for all integral values of vi 
and n, positive or negative. 

We might have assumed cos x = z instead of sin x = z. 

This method is applicable to any rational trigonometric 
differential. 



EXAMPLES, 
cos X / sin^ X sin-' x sin x " 



/• . , „ , cos x/ sin" X snr'x sinx\ , x 
1. I sni-txcos-!x(Zx=^^(^-^j ^.^ S") To' ^^ 

Let sin x = 2 ; .-. cos-x = 1 — 2-, dx = (1 — z-)- ^ ''- dz. 

_ (1 — Z-y ''- /Z^ _ t:^ _i\ y •^'"' 

~ 2 \ 3 12 S ^ 

Sulistituting sin.r, for .;, wo obtain tlio integral in \\\. 



+ „; 



200 INTEGRAL CALCULUS. 

2. I aec^xdx = —J — (sec^ ^ + 9 sec ^) + q log (sec x + tan x). 

dz 



zVz-2-1 



Let sec x = z, or x = sec— 1 z ; .-. dx = 

= 1(^2 _ 1)1/2(^3 + 1^) + |l0g(2+V^2_l). 

Substituting sec x for z, we obtain the required integral. 

/^cos^xdx cos^x , , , X 

3. I — ^ = — h cos X + log tan - • 

I smx 3 ^2 

Assume cos x = z. 



. /* dx cos X , 1 , ^ X 

'*• I • q = — f, ■ ., — h - log tan - 

J sm^x 2 sm-x 2 * 2 



^ /» ■ C?X 1 , . ^ X 



sin X cos2 X cos x 2 



dx _ /^ (sin^x + cos2x)dx 
sin X cos^ X 



6. r . '^ ^ = r 

^ sm X cos'^ X ^ 

/ sinxdx /*_ 
cos^x J si 



'sinxdx , /* dx 



sm X cos^ X 



„ /• dx 1 cos X , 3 , ^ X 



sin^xcos^x cosx 2 sin'^x 2 2 



— m — n = I ro + ., I , ., log {a cose + b sm ^). 
a + & tan d a^ + 62 q^2 4. 52 » ^ / 

Assume tan 6 = z. 



-/jsif^=i'--a-0-r 



TRIGONOMETRIC FORMS. 201 

. « >. rr. C • -, -, e^^ (a sin bx — b cos bx) 
194. To prove \ e^^ sm bx dx = ^ ^ ^ ; (1) 

C T -, e*^(bsinbxH-a cosbx) 
and I e^^ cos bx dx = ^ ^ ^ ^ • (2) 

Integrating e"^ sin hx dx by parts, first with u = sin hx and 
then with v. = e^"^, we obtain (3) and (4). 

/. , , e""^ sin ^.x /> /* , , 

e«^ sm occ ^ic = I e'*'= cos 6u3 dx, (3) 

/. , , e"""^ cos hx , a C , , 

e"^ sm hxdx = ^ 1 ) ^"^ ^^^ ^^ ^^- (^) 

Subtracting (3) from (4), we obtain (2). Multiplying (3) 
hj a/b and (4) by b/a, and adding the results, we obtain (1). 

EXAMPLES. 

1. I e«^ (sin ax + cos ax) dx = e«^ sin ax /a. 

2. j e3^(sin2x — cos2cc)dx = e3^(sin2a: — 5 cos2x)/13. 

/'sin jc , sin X + cos x 

3. i dx = 

J e^' 2 e^ 

/^ sill — ^ T f^T 

4. I — —777, = z tan 2 + Iog: cos z, where ^ = sin- ^ x. 



/ sni— 'xda 

/sin-ixdx /* ,, , ^ r^ 

— — r— ; = I 2; • ^o.c-zdz = 2 (an -i — I tan z 



(. :^x + sm X , . ^ T> 4. o 

5. I T-, dx = x tan - • Put x = 2z 

+ cos X 2 



195. Integration by expansion in series. When by any 
of the preceding methods we cannot integrate a given differ- 
ential exactly, we can expand the differential in a series, 
integrate its terms separately, and thus obtain the integral 
approximately between the limits of eonvergenev of the 
series. 



202 INTEGRAL CALCULUS. 



/ 



EXAMPLES. 

x^ , x^ x' , x^ 



Multiplying by dx/x and integrating, we obtain (1). 

/»cosa3, , 0^2 X* x^ , x^ 

2- J ^^''^ = '°S ^ - 2:12+444- 6^6 + H8 

e«^ , , , , a2x2 , a^x^ , a^^i 

dx = log X + ax + ^ + ^ + ^ + • 



/^ 



(1) 



C—^^—= 1 • x^ 1 • 8 • x^ 1 • 3 • 5 ■ xi3 
* J Vl +x4 ~^ 2-52-4-9 2- 4-6. 1.3 

_ ri logx - / 1 , 1 , 1 , 1 , 

5- j^ r±^'^-=-(p + p + 3^ + 45+- 

(1 — x)-i = 1 + X + x2 + x3 + • • •, T^•hen — 1 < x < 1 ; 

J^ilo£x^^_ r (iog2. + 2.iog3, 4.a,2iogx+ • • •)cZx. (2) 

-*- ^ Jq 

Integrating each term in (2) by the definite integral given in 
example 2 of § 179, we obtain (1). 

6. By integrating (1 + x"-)— ^ dx directly and by series, prove that 

tan-ix = x-| + f-^ + f , (1) 



(1 + X2)-1 = 1 — X2 + X* — X6 + X8 — • • • : 
r^dX pn _ 3.2 + 3^4 _ 3^6 + 2^8 _ 

Jo 1+^' Jo 

^ ^ . . , x3 , 1 • 3 • x5 1 • .3 • 5 • x" , 

7. Prove sni-i x — x + -— r + - — -. — - + ^^ . ,. ^ + 
2-3 2 ■ 4 • o 2 • 4 • 6 • / 



')dx. 



Integrate dx/ V 1 — x^ both directly and by series. 

8. Prove log(a + »:) = loga + | - ^. + ^s " l| + 

Integrate dx/ [a + x) both directly and by series. 



1 • x3 1 . 3 • x5 1 • 3 . 5 • x7 



9. Prove log(x+Vl+x2) = x--^ + -^^^^-^^^^^ + 



CHAPTER VII. 

LENGTHS AND AREAS OF CURVES. SURFACES AND VOLUMES 
OF SOLIDS OF REVOLUTION. 

196. Lengths of curves. Bectangidav co-ordinates. Let 
s denote the length of the arc whose ends are the points 
(^oj 3/0) ^^^ (^j y) '1 then from ds^ = dx^ + dy^^ by § 165, we have 



^0 
or 



5= C\\^{dyldxf'f'''dx, (1) 

s=C\(dx/dyY + lJ^-'dy, (2) 



according as we express ds in terms of x or of y. 

In any given curve we use that formula which gives the 
simpler expression to integrate. 

EXAMPLES. 

1. Find s of the semi-cubical parabola ay'^ = x^. 
{dy/dxy^ = 9x/4a; 

=if[(-FT-(-^'n- <^) 

When (xo, yo) is the origin, (1) becomes 



=iy[(-e)"-] 



2. Find s of the cycloid x = r vers-i (y/r) qi V*J ry — y'-. 
{dx/iW = y/{2r-y); 

.'. s = V27- I (2 r - //)- 1 / - dy § 1 Ot?, (2) 

= 2 V27- [(2 r - yoY '- - (2 r -yY'-]. 



204 INTEGRAL CALCULUS. 

Putting ?/o = and ij — lr and taking twice the result, we find 
that the length of one arch is 8 r. 

3. Find s of the parabola 2/^ = 4_px, (xo, ?/o) being the origin. 



y I 11 -\- V 4 xp' 

Ans. s = -p-V4p2 4- ^2 -f- p log ^ 



4p -f o 2p 

4. Find s of the circle x^ + ?/2 — ,.2^ ^nd the circumference. 

Ans. r sin— 1 (x/r) — r sin— i (xo/r) ; 2 7rr. 

5. Find s of the hypocycloid x^/^ + ^2/3 — (x2/3^ ^^d the length of 
the curve. 3 1 /q/ o/q 9 /q^ o 

6. Find s of the ellipse y^ = (1 — e^) (a^ — x2), and the length of curve, 
e being the eccentricity. 



C\a2-e2x^y/2 ^_Z__ ; (1) 



dx 
-0 Va2 - x2 ' 

hence, the length of the elliptic quadrant Sq is 



{a^-e^x^)^y^-==- (2) 

Va2 _ x2 

The integrals in (1) and (2) cannot be obtained directly, but 
(a2 — 62^.2^1/2 can be expanded by the binomial theorem, and the 
terms of the result can be integrated separately. Thus, 

J"^ « dx ^ /^« xJ^dx e^ /^" x^c?x 
Va2 _ x2 2ajQ Va2 - x2 Sa^J^ Va2 _ 3^2 

_ 7m / e2 3 • e^ 32 • 5 • e^ _ _ \ 
~ 2 V 22 22.42 22.42.62 ' ' ')' 

7. Find s of the ellipse when given by the equations, 
X = a sin ^, y = b cos 6, 
where 6 denotes the complement of the eccentric angle. 
dx2 = a2 cos2 ede^, dy^ = b^ sin2 edd^-, 
.-. ds2 = (a2 cos2 ^ + 62 sin2 e) dd^ 
= a2 (1 - e2 sin2 ^) d&K 

.'. s = a C (1 — e2 sin2 e)^'-dd; 



J ^77/2 1 



— ^ e2 sin2 ^ — - e* sin* d — — e^ sinS ^ )dd. 

2i o Id 



THE CATENARY. 



205 




~7a 



197. To find the length and the equation of the catenary. 

Let NOM be the curve in which a chain or flexible string 
hangs when suspended from two fixed points If and N\ then 
NOMi^ a catenary. ^ 

Let w denote the weight 
of a unit length of the chain, 
and s the length of the arc 
whose ends are the lowest 
point (0, 0) and the point 
(x, y), or B ; then the load 
suspended, or the vertical 
tension, at ^ is siu. Denote the horizontal tension, which is 
the same at all points, by aw. Let DA be a tangent at B ; 
then if c • BD represents the total tension of the chain at B, 
cBE and cED will represent, respectively, its horizontal 
and its vertical tension at B. 

dy _ c- ED 
dx 



O' 



X' 



Hence, 



BE 



siv 
aw 



(1) 



Vc/s^ — dx^ 



dx 



.'.dx 



ads 



^a- + 



. . x = a ) 



ds 



= a Iol;- 



.9+V<r2 + ^.2 



Solving (2) for ,9, we obtain as tlie length of OB 
Eliminating s between (1) and (3), wo obtain 



(2) 



(3) 



. y-\-a =-(^r'/" + < 



^■i) 



is the equation of the catenary rofornnl to the a\t\^ ()A' and ( > Y. 
If 0'0 = a, and the cnrvc be rc^lVM-rml {o the axes (>'A' and 
O'Y, its equation will evidently be 



206 



INTEGRAL CALCULUS. 



198. To find the length and the equation of the tractrix. 
The characteristic property of the tractrix is that the length 




T X 

of its tangent FT is constant. 

Denote the constant length of the tangent PT by a. 

Let PM=ds', 

then — PN = dy, and JSfM = dx. 

ds__ _P^_ _^ 
' ' dij~ ~ P]Y~ ~~y 

Hence, if s is measured from A^ or (0, a), we have 



C'dy , a 

— — a \ -^ = a log 



y 

Again, from the figure we have 



y 



(1) 

(2) 

(3) 



dy /dx = — y l-\/ a- — y'^. 
Integrating (3), remembering that ?/ = a when a: = 0, we have 
x = — ^ a} — y'^ + a log [(fl^ + Va^ — 2/^) /?/] 
as the equation of the tractrix. 



199. Lengths of polar curves. Let s denote the length 
of the arc whose ends are (/oq, ^o) ^'^^ (p? ^)5 ^^^^ from 
ds = V^WT^, by § 165, we have 



s= r^p\W^-\-dp% or s= f'VpWT~dp 
according as ds is expressed in terms of B or p. 



(1) 



CURVES IN SPACE. 20^ 



EXAMPLES. 
1. Find s of the spiral of Archimedes p = ad. 



ds = aVl + ff^dd; 
'.s = a I VlTV^dd 

= ^ [dVT+J' + log {6 + Vi + e')~] ^• 



Putting ^0 — and 6 = 27t, we obtain as the length of the first 
spire 

a [ttVi +i7f^ + |log (2 TT + Vl +4 7r2)]. 

2. Find s of the logarithmic spiral p = he^/". 

6 /a = \og{p/b) = log p — log 6; 
.-. pdd = adp. 



.-. ds = V a- + 1 dp. 



'^X 



p 
dp 

Pn 



= Va-'+ 1 (p-po). 

3. Find s of the cardioid p = 2 a (1 — cos 6). 

Ans. s = 8rt[cos(^o/2) — cos(^/2)] ; entire length = 16 a. 

4. The entire length of the curve p = a sin^ (d/o) is 3 7ra/2. 

200. Curves in space. Let s denote the length of an arc 
of a curve in space wliose ends are (.r,,, //„, .-„) '^^^'^^ (•'"' .'/• -)> 
and let As denote the length of an infinitesimal arc whose 
ends are (.x, y, r:) and (.r + A.r, // -h A//, -' + Av"); then 

A.s- = VA.r"-^ + A// + A.r- + ri", wlunv ;/ > 1 . >J (W) 

.-. ds = V(/x2 + dtr + </--. § 7 1 



.••*•= r V(/.r- 4- df/- + </-'. 



208 



INTEGEAL CALCULUS. 



EXAMPLES. 
1. Find the length of an arc of the helix, 

x = a cos 6, y = a sin d, z = kd. 
Here dx = — a sin 6dd, dy = a cos 6dd, dz = kdd ; 



ds = VoM^ de. 
; s = Va2 + k-^ j dd 



2. Eind s of the curve y = x^/2 a, z = x^/6a^, s being reckoned from 
the origin. 

r-2 . 

dx 



S.V^A 



= x -\- —^-, = X + z. 

3. Find s of the curve y = 2 Vox — x, z = x — (2/3) Vx^/a, s being 
reckoned from the origin. 



"X'(Va'^Vx-0^^^^ + ^-"- 



201. Areas of curves. Let JYBC be the locus of y = ^x, 

, and JVDSC that of y = fx. 
Denote their intersections, J^ 
by (^0, 2/o) and C by (a:i, t/i), 
and the variable area NBD 
by A 

Let X = Oilf, c?ic = DE', then 
c?^ = DBLE = {^x - fx) dx. 

,'.A'= r\^x-fx)dx, (1) 




o M 



Xq 

where A 



Writing the equations of the curves in the form 
x = (f>-^y and x=f-^y, 



AREAS OF CURVES. 209 

in like manner we obtain 

A'=r\f-'y-<j.-hj)d,j. (2) 

If the locus of y =fx is the cc-axis, formula (1) will become 
identical with (o) in § 16G. 

EXAMPLES. 

1. Find the area bounded by the parabolas ?/2 = 4 ax and x^ = 4 ay. 

The parabolas intersect at the points (0, 0) and (4 a, 4 a) ; hence, 
the limits are x = and x = 4 a. 

The curves enclosing the area are y = V4 ax ^^^^ V ~ x-'/4 a. 



X"( 



4: ax — -T— ) dx, = — r— 
4a/ 3 



2. Find the area bounded by the parabola x^ = 4 ay and the witch 
2/ (x2 + 4 a'-^) = 8 a:\ 

A r''"/ 8a3 x2\ ^ /^ 4 

Ans. ^ = I ( , , , — ; — T— ) dx = a- ( 2 TT 

J_^,^^\x- + 4«- 4 a/ \ -5 

3. Find the area bounded by the curve y (1 + x-) = x, and tlie riuht 
line y = x/4. ^,i.s.. loo- 4-3/-!. 

4. Find the area of that part of the ellipse x-/a- + i/-/l>'- ~ 1< which 
lies between the lines x = Xq and x = Xi. 

Here the boundiui;- curves arc 



y = {h/a) Va- — x-, y = — {h / a) V((- — x-. 



'' r A"i ; , .. • , •'■"1 '"1 

= - I X \ a- — X- + a- sm— ' • 



(n 



Putting Xo = —a and Xi ■ " a in (i). wc i^btain rrnh as tlu> area ^>t■ 
the given ellipse. 

Tutling h - (I in (1), wi' itbtain [he -avca of (liat part oi tl\o circle 
j^-2 + ?/- -; (('-' Nvhii'h licij bc(\\i-cn the lines x - .)\i. x .r,. 



210 INTEGRAL CALCULUS. 

5. Find the area of tlie hyperbola x'-/a- — y'- /IP- = 1, included 
between the lines x = Xq and x = Xi. 

h 



Ans. - X Vx^ — a- — a- log (x + Vx^ — a^)\ • 



6. Find the area bounded by the catenary, the x-axis, the ?/-axis, 
and any ordinate y. 






(e-x/a -I- e- ■'■/«) fZx = as. § 197, (3) 



7. Find the area bounded by the tractrix in the first quadrant, the 
X-axis, and the ?/-axis. 



Jydx = — I Va- — y'^dy = 7ra'^/4. 
^ a 



8. Find the area between the cissoid y'^ (2 a — x) = x^ and its 
asymptote. j^^s. Syta^ 

9. The area of the hypocycloid x-/^ + y-^^ = a-''^ is 3 ita- f^. 

10. The area of one loop of a-?/-* = x^ (a- — xP-) is 4 a-/ 5. 

11. The area of one loop of aHj- — W-x- (a- — xP-) is 2 ah /Z. 

12. Solve the first example by formula (2) in § 201. 

13. Find the area between the curve y'-x — 4 cC- (2 a — x) and its 
asymptote x = 0. j^^^s, 4 ^(■^2^ 

202. Areas of polar curves. Let B be any fixed point 
(p^, 0,^ and F any variable point (p, 6). Conceive tire area 
^ BOP as generated by tlie radius vec- 
tor p, and denote it by A. 
Dyj^ With OB as a radius draw arc PD, 

and let clB = ABOB' ; then 

^^= 0BB = {/I2)dd', 

1 Z*^ 
.•.^ = i p2(^^. (1) 

For the proof of (1) by limits see § 71, example 11. 




AKEAS OF POLAR CURVES. 



211 



EXAMPLES. 
1. Find the area of the cardioid p = 2 a {1 — cos d). 

dd = 6 Ttofi. 



J ^277 
(1 — COS d)'^ < 



2. Find the area of the lemniscate p'^ = a^ cos 2 



Area = 2 






cos 2 6d9 = Or'. 



3. Find the area between tlie first and the second spire of the spiral 
of Archimedes p = ad. 



Area 



iC'"-'j: 



e-2 de = s a^7t^. 



4. The area generated by tlie radius vector of the logarithiuic spiral 
p = e«^ from ^ = to 6* = 7r/2 is (e'^" — 1) /4 a. 

5. The area of one loop of the curve p = a sin 2 6 is 7ta~/8. 

6. The area of one loop of the curve p = a sin 3 d is 7ra-/12. 

7. The area of a sector of the spiral pO = a is (6 — 6^) (f-/2 6^^). 

n'2 ft 

8. The area of a sector of the spiral p-d = a- is — log — • 

2 ^0 

9. The whole area of the curve p = a cos 2 6^ is Tra-/^- 



203. Areas of surfaces of revolution. Let A be a tixeil 
point (.-^Co, ?/o) and P a variable point (.r, i/) on the curve niAP. 
Let ^P = .9, and PP' = A.s'. Draw ^y 

FT and P'P each parallel to OX y 
and equal to As. Let S denote the 
surface generated by the rovohitiou 
of AF about the .r-axis ; tlien A.S^ '" 
equals the surface generated by P7*'. 

Evid(^ntly 



Q 



M N 



that is, 



surface /' 7' < AN < surface P'R; 
'2 7r//As < AX < '2 it (^// -f A//^ A.- 



212 



INTEGRAL CALCULUS. 



Let As — i; 

then AS — 2 tt^As + vi", wliere 7i > 1. 

.'. dS = 2 iryds ; ov S = 2 tt j yds. 



(1) 



In any particular example, yds is obtained in terms of x, y, 
or any other variable as may happen to be convenient. 
Similarly, if the ?/-axis is the axis of revolution, we have 



? = 2 TT / j'ds. 



(2) 



204. Volumes of solids of revolution. Let ^ be a fixed 
point (Xq, 2/o) and P a variable point 
(x, y). Let V denote the volume 
of the solid generated by revolving 
Pv4PJ/ about OX. 

Conceive this solid as generated 
by a circle whose centre moves 
along OX, and whose variable 
radius is the ordinate y of the 



- 


1^-^ 


u 


D 



O B 



M N X 



curve AP. 

When X = OM let dx - JP¥; then 

d V = cylinder MPDN = iryhlx. § 11 

.-. F-TT f%/dx. (1) 

Similarly, when the y-Sixis, is the axis of revolution, we 
have 



V=7r 






(2) 



The proofs of (1) and (2) by the method of hmits are left as exercises 
for the reader. 

Li the followmg examples, a segment of a solid of revolution means 
the portion included between two planes perpendicular to its axis ; and a 
zone means the convex surface of a seorment. 



SURFACES OF REVOLUTION. 213 



EXAMPLES. 
1. Find the area of a zone of a sphere. 
Here iids ~ rdx. 



= 2 TC I yds = 2 7rr j ( 



.: S = 2 7C I yds = 2 vrr i dx 

•Jxq 

2 7ir(x — Xq). 

2 [..„];= 



The entire surface = 22 Tirx = 4 jtr'^ 



2. Find the area of a zone of the surface generated by the cycloid 
revolving about its base. 

yds = v2 r (2r — y)~ ^'-ydy. 

.'. S = 2 7fV2r I ''y (2 r - y)~ i /'-' dy 

= 2 7tV2r[— ^ (4 r + ?/)(2 r — ?/)i /^~\ ''■ 
L ^ J //o 

The entire surface = 4 tt V2r — ^ (4 r + y) (2 r — ?/)i ^- 

-64 7rrV3. 

3. Find the area of a zone of a prolate spheroid. 

The generating curve is y- = (1 — c ') {a- — x-) and 



?>,/-,- 



yds = - \a- — e-x'-dx. 
.'. S = 2 7t~ / \(i^ — 



c-x- dx 



TT 



'V\\c. (nitir(«, surface -: 2 7rb\l) -f- ((f/r)siu ' c]. 

4. Find (he area of a /.one of (ho surface generated by the c;Ucn;u\v 
revohing about the .r.-a\is. 

Here yds -- [c' /" + c •' /"Y-dx. 



... ,S = n p^"- (c^'-'^o - t--'-' /-)■ + tul ' • 



214 INTEGEAL CALCULUS. 

5. Find tlie area of a zone of the surface generated by the tractrix 
revolving about the x-axis. Ans. 2 7ta (vq v). 

6. Find the area of a zone of the paraboloid of revolution. 

Ans. -^ [(4p2 + ^n - (4^2 + y^^fn-^, 

7. The entire surface generated by revolving the hypocycloid 
3^2/3 j^ y-2/z — q2/s about the x-axis is 12 no?- /h. 

8. The surface generated by revolving the catenary about the ?/-axis, 
from X = to X = a, is 2 ita?- (1 — e— i). 

9. Find the volume of a segment of the prolate spheroid. 

V = 7t I y'^dx = 7t— j (ci? — X-) dx 

J X(^ J Xq 

a- L ^Axq 

52 p x^'~\ ^ 4 

The entire volume = tt- - a-x — v ~ q italfi, 

which is two-thirds of the circumscribed cylinder of revolution. 

Putting h = a we obtain the volume of a segment and the entire 
volume of a sphere whose radius is a. 

10. The volume of the oblate spheroid is two-thirds that of the circum- 
scribed cylinder of revolution. 

11. The volume of the paraboloid is one-half the circumscribed 
cylinder of revolution. 

12. The volume of the solid generated by revolving an arch of the 
cycloid about its base is five-eighths of the circumscribed cylinder. 



Here V=2.r^MK=- 

Jo V2ry — ?/2 



13. Find the volume of the solid generated by the revolution of the 
tractrix about the x-axis. 

Volume = TT / y-dx = — 7t I Va- — y- ydy = no?'/ 3. 
»/ J a 

14. The entire volume generated by revolving the hypocycloid 
a;2/3 + ^2/3 = ci2/3 about the x-axis is 32 7raV105. 



VOLUMES OF SOLIDS. 



215 



15. Find the volume of a segment of the solid generated by the revo- 
lution of the curve / (x, y) = about the line x = a. 

Let AB be the line x = a, and let P 
be any point on the curve /(x, y) = 0, 
or EP ; then AH = x — a. 

Let BC = Ay = i ; then 

AV = 7t{x — a)2 Ay + ui«, 
v^here n> 1. 

.-. F = TT r"{x - ay- dy. (1) 

The student should prove (1) by the method of § 204. 

16. If the figure bounded by x = a and the parabola ?/2 = 4_px is 
revolved about the line x = a as an axis, the volume of the solid gener- 
ated is 32 Tta^ Vpa / 15. 

17. Find the volume of the solid generated by the revolution of the 
cissoid about its asymptote. Ans. 2 7i-d^. 




205. Let V denote the volume generated by any plane 
figure moving parallel to a fixed plane. Let a* denote the 
distance of the generating figure from some fixed point, and 
let <^ic denote its area ; then, evidently, 

A V lies between <^ (.r) • \x and </> (,r + A.r) • A.r. 
Hence, when A.x = i, 

AV=(f> (.>■) ■ A.r + rl", wliere // > 1. 

the limits being so cIioscmi as to includo the volume sought. 



EXAMPLES. 

1. Kind the volume oi any pyi'auiid (>r cone. 

Let J> denote the area of ihi> basi> anil a the ahitudo. 

Let 0x denote the area of a stn-tion parallel to thv' base at the 



distaiUH' ,(• fr(>in tlu* vertex. TIumi bv 



rv we ha\e 



0.1' : /) — x'~ : a- 



i.r — lW-/a'-. 



216 



INTEGRAL CALCULUS. 



Conceive the solid as generated by tliis variable section moving 
from the vertex to the base ; then by (1) of § 205 we have 



V={B/ 



a'^) r"x'-i 



■dx = B ■ a/S. 



2. Find the volume of a right conoid with circular base, the radius of 
the base being r, and the altitude a. 

Conceive the solid as generated by the 
section BTQ moving to the right, and 
denote OP its perpendicular distance 
from by x. 

OC = AB = 2r, 
OA = CB = a. 
.'. (px = PQX PT 




= a\2rx — x'^. 



'/:■ 



V = a I V 2 rx — x'^ dx 
■7tr^a/2. 



3. A rectangle moves parallel to and from a fixed plane, one side vary- 
ing as its distance from this plane, and the other as the cube of this dis- 
tance. At the distance of 3 feet the rectangle becomes a square of 4 feet. 
Find the volume then generated. Ans. 9^- cubic feet. 



4. An isosceles triangle moves perpendicular to the plane of the ellipse 
x2/a2 + ?/2/62 = 1, its base is the double ordinate of the ellipse, and its 
vertical angle 2 A is constant. Find the volume generated by the triangle. 

Ans. 4 ab'^ cot A / S. 

5. A woodman fells a tree 2 feet in diameter, cutting halfway through 
from each side. The lower face of each cut is horizontal, and the upper 
face makes an angle of 45° with the horizontal. How much wood does 
the man cut out ? jins. 4/3 cubic feet. 

6. Obtain formula (1) in § 205 by the method of proof employed in 
§204. 



CHAPTER VIII. 
DOUBLE AND TRIPLE INTEGRATION. APPLICATIONS. 

206. Double and triple integrals. If we reverse the oper- 
ations represented by - — —dxdy, we obtain the function u. 

it Ju Ct U 

That is, "=//|l;^'^^'^' (1) 

which indicates two successive integrations, the first with 
reference to ?/, x and dx being regarded as constants, and the 
second with reference to x, y being regarded as a constant. 

In (1) the right-hand sign of integration is used with the 
variable 2/; that is, the signs of integration are taken from 
right to left in the same order as the differentials. 

Let h' denote the definite integral when the limits for x are 
iCo and iCj, and those for y are ?/„ and y^ ; then 

Ex. 1 . r " C\i){x- y) dx dy = C "x dx \~ ~ '!f\ ' 

J^"/x~b- xb-^\ , 

= a-l)-{i( -/')/('>. 
Oftentimes the limits of tlie first, integration are funi'tions 
of the variaWe of tlie second. 

1.x. 2. j j (x + //) d!,dx = j {Wf^+ I.- -r ) ''-'^ " -JO ■ 

The second nuMubcM- of (H lUMioti^s what is calUHl an itideji- 
nlte double 'uifajral, and the seci^ml humuIht ol" \^1^ a drfhntc 



218 INTEGRAL CALCULUS. 

double integral. Similarly, we have indefinite and definite 
triple and multiple integrals. 



Jr*2a /*x r*x r*2a r*x r'x 

I I xyz dxdydz=i i xy dx dy I zdz 

X'^^xdx C , ^ .,, , 
-y- I y{x^-y")dy 



'2a X5 , 21 a6 



EXAMPLES. 



1. I I ydxdy = — 



•/o •/o 

*/ 



/32 sin (pdpd(p = — 
o 



xydxdy = — 
^ 



J'^b /'p/b 752 

pdpdd = -^ 
6/2 */0 



24 



. r^ ^22/ 11 &4 

5. I I xydydx = -^ 



j pdcpdp = -{a'^-b'). 

*/2 b cos <|> 



2 

a Jfi 

ab^ + 2 a4 



Xf^ /'y 6^ — a=^ 

I p2 sin edpdd = — - — (cos /3 — cos 7; 

8. I I (x — a) (y — b) dy dx = a^b 



3 
16 



I I {y'^^z''-)'kdzdydx-—koP. 

-a *y — a *y — a 

Xb /*a /•26 1 

I I x'^y^z dx dydz = - aW {W — a^) 

Jr*2 f»x /*x + y 
I I e^+y + ^dxdy 

«/o •/o 



6^-3 3e4 , „ 
dz = —^ -4- + ^'- 



AREAS BY DOUBLE INTEGRATION. 



219 



Rectangular co-ordi- 

Ic 



207. Areas by double integration 

nate. Let NBC be the locus 
of ?/ = cfix, and NDSC that of 
y = fx. Denote their inter- 
sections, N by (xq, ?/o) and C 
by (xi, yi), and the curvilinear 
area NBCD by A. Let P be 
any point (x, y) in this area, 
X and y being independent. 

Let PP= dx and P(^ = ^?/. 

When X and rfcc are con- 
stants, P (t ()P, or dx dy, will be 
the differential of the area DPFE. Hence, integrating dx dy 
between the limits MD and MB, or fx and <^x, we obtain the 
area DBLE, or (<^cc —fx)dx, which is the differential of the 
area NDB. Integrating {c\>x — fx) dx between the limits Xq 
and Xi, we obtain the area NBCD, or A. 




O M' 



Hence, 



I dx dy 



(1) 



When y and dy are constants, PGQF, or dydx, will be tlie 
differential of the area HPGK. Hence, integrating dydx 
between the limits //'// and H'S, we obtain the area IISRK, 
which is the differential of the area JSUIS. Integrating tliis 
between the limits M'JV and XC, we obtain the area XBCJ>, 
as before. 



Hence, 



-/= f f<r'/<^'^; 



(2) 



tlu^ limits being taken so as to iiu'liule the V(\]mvvd area. 

The ()i-(Un- of integratiiMi, therefore, is iiulift'erent, j^roviiU^l 
the limits assigned in each i-asi^ be siu'h as to ineliule the 
area souglit. 



Cor. 3',,/./ = (/xdy and 9^^, .1 = dydx. 



220 INTEGRAL CALCULUS. 

Ex. 1. Find the area bounded by the parabolas y- = 4 ax and x- — 4 ay. 
The parabolas intersect at the points (0, 0) and (4(7. 4 a). 
Hence, if we nse formula (1), the constant limits for x will be and 
4 a, and the variable limits for y will be x-/4 a and V4 ax. 



That is, area = { | dxdy — ^-;^ 

V4a '^ 



Using formula (2), we obtain 



area =1 I dy dx = — -7- 



-/4. 

Ex. 2. Find the area between the parabola y- = ax and the circle 



. > i i --"'"■' , , ;ra- 4o- 



208. Areas of polar curves by double integration. Let 

XBG be tlie locus of p = 0^. and liX'i'' tliat of p =fO. 




Let Z :^0X= 0,, and Z A^OG = 6^. 

Denote the area HEGX by A. 

Let P be any point (p, ^) in this area, p and 6 being 
independent. 

Let PJ/ = dp and ZPOS= dO ; 

then arc P,S' = pdB. 

Construct the rectangle PCA2I wdiere FC=-FS = pdB\ 
then triande POC= sector P 0/S. 



AREAS BY DOUBLE INTEGRATION. 221 

When 6 and dO are constant, FCAM, or pdOdp, will be the 
differential of the triangle FOC, or of its equal, FOS. Hence, 
integrating pdddp between the limits OD and OB, or fO and 
<^e, we obtain the area DBB'D\ or ^ \_{<iiBf — {fOf^ d6, which 
is the differential of the area FLDBN. 

Integrating this differential between the limits ^,j and ^i, 
we obtain the area HEGN, or A. 

Hence, A= I ' I pdOdp. (1) 

Coil. BopA = pdOdp. 

EXAMPLES. 

1. Find the area between the two tangent circles p = 2a cos d and 
p = 2b cos 6, where a >► b. 

J'*rr/2 r*2acose 
I pdOdp 

) */ 2 6 cos 9 

X7r/2 
coii^edd= 7t{a^ — l)^). 

2. Find the area, (1) between the first and the second spire of the 
spiral of Archimedes p = ad ; (2) between any two consecutive spires. 

3. By double integration find the area, (1) of a rectangle; (2) of a 
parallelogram ; (.']) of a triangle. 

4. Find the whole area of the curve (?/ — mx — r)- = a~ — .r-. 

A)is. rra'-. 



209. Area of any surface by double integration. (>ii tho 
surface rj — _/'(.r, //), let /* be any point (,r. //. ,:). and (.-' {\\c 
])(nut (.r + A.r, // -|- A//, ,*: -f A.:), .r ;uul // biding indi^pcndcnt ; 
tlien F'N= Axiiml /".)/ A//. 

Conciuve a tangmit. pl;nu> at. /*. not shown in the tigiir(\ 
'V\\o planes throngli /'and (^) parallel to th(^ eo-in-dinatt^ plant\s 
A'Z and )'/ will cut a. enrv(>(l (piadrilateral /'(,> I'roni the sur- 
face ,~=7'(.c. //), and a iKirallelegraui /'./ from the tangent plane. 



222 



INTEGRAL CALCULUS. 



Let Ace = i and Ay = vi ; 

then area FQ = area Pq -\- vi^, where n'> 2 

= area F'Q' • sec y + vi'', (1) 

where y is the angle which the tangent plane at P makes 
with the plane XY. 
From (1) we have 

A^.,^S = Ax Ay sec y + vi^. 
.'. 3^yS = sec y • dx dy. § 140, Cor. 




From 


analytic geometry we have 






>~'-[' + (l)'^ 


■(I)'] 




■■■-//[-(I 


on* 



] 



dxdy, (2) 



the limits being so chosen as to include the required surface. 
• Let S denote that part of the surface z =^f(x, y), z being a 
one-valued function, which is included by 

the cylindrical surfaces y = <j)oX, y = cfiX, 

and the planes x = a, x = b -, 



AREAS BY DOUBLE INTEGRATION. 



223 



then 



S 



rx:[ 



dxj \ dij 



)2 -|l/2 
J dxdy. (2) 



210. Volume of any solid by triple integration. 

Let P be any point (ic, ?/, z) within the solid OZY-X, x, y, 
and z being independent. 

Let PS' = dx, PS = dy, PP' = dz. 




Regarding it", dx, y, and dy as constants, the prism FK\ or 
dxdydz, will be the differential of the prism NK. Hence, 
integrating dxdydz between the limits z = and z = XH, wc 
obtain NEdxdy, or the prism NE', which is the diftVrtMitial 
of the solid MM'A'A-P. 

Integrating NRdxdy between the limits // = and // = .!//>, 
we obtain the cylinder MAB-1V, or MMUli\ wliich is the dif- 
ferential of the solid OZY-]\r. 

Integrating MABdx between the limits x and x OX, 
we obtain the volume OZY-X, or ]\ 

Hence, /'=- C C Cdxdydz, 



0^ 



tlie limits beini;" so chostMi as to iiu'huh> tht^ viduim^ smiulit. 



224 INTEGRAL CALCULUS. 

Let V denote the volume bounded by 

the curved surfaces ^ =/o (x, y), - —f{^^ V) 5 

the cylindrical surfaces y = ^qX, 7/ = cf>x; 

and the planes x = a, x = b ; 

I j dxdijdz. (2) 

Cor. 9x^2^'^= dxdydz, 9y,a;F= dydzdx, • • •. 

Ex. Eind tlie volume of the ellipsoid x'^ / a- + y- /h- -\- z"- / (S^ — \. 

The entire volume is eight times that in the first octant, where the 
limits are 



••• T 



2 = 0, z — c Vl — a:-/a- — y- /l>^- ; 
2/ = 0, ?/ = 6V1 — x-/a^; 
aj = 0, X — a. 






EXAMPLES. 

1. Find the volume bounded by the plane x — a and the surface 
2- / c + ?/"-^ / 6 = 2 X. 

The entire volume is four times that in the first octant ; 
.-. F=4 I I I dxdydz = Tta-^bc. 

2. Eind the volume bounded by the surfaces, 

x2 + y2 =z cz, x^ + ?/- = ax, 2 = 0. 

dxdydz ~ 



^0 »/o 



32 c 



3. Eind the volume bounded by the cylinder x- -\- y'^ = r^ and the 
planes 2 = and 2 = 7/ix. Ans. 4mr3/3. 

4. Eind the volume bounded by the surface x%- + a'^y- = c^x^ and the 
planes X = and X = a. j^^s. 7tc^a/2. 



VOLUMES BY DOUBLE INTEGRATION. 225 

5. Find the area of the zone of the sphere, 

x2 + y'^ + z'^ = r\ (1) 

included between the planes x = a and x~h. 
From (1), Qz/dx=—x/z^ 'dz/(ly — —y/z. 

Hence, [i + (gg)% (-Q^V T" = , ^ ■ 

L \ ax; \dy J J Vr^ — x"^ — y^ 

The area required is four times that in the first octant, where the 

limits are x = a^ x = b, y = 0, y = Vr'-^ — x^ ; 



.•.area=r4 f f '''"' ^ '^'^'^^ ^ = 2 7rr(6 - a). § 209, (2) 

Ja Jo ^r- — x^ — y'^ 



6, Find the surface of the cylinder x'^ + z- = r- intercepted by the 
cylinder x^ + ?/2 = 7-2. Ans. 8 r^. 

211. Solids of revolution. Let P be any point (x, ?/) in 
the area NBCD (§ 207, fig.), x and ij being independent. 

Let PF = Ax and FG = Ay. 

Conceive NBCD to revolve through radians abont OX as 
an axis ; then 

0// -AxAy< a].,^ V < (// + A//) • Ax Ay. 
Hence, when Ax = i, and Ay = vi, 

A^.,/F= OyAxAy + ?»/", where n > 2. 
.■.^,,V=6ydxdy. 

.-. F=^ P r%^/-r.///. (1) 

Putting = 2 TT, we obtain tlie vohmie generated by a com- 
plete revolution of the area. 

CoK. If the o-'axis cuts the area, formula (\) will giv^^ the 
difference between the volumes generated by tlu> two parts. 
Hence, V =0 Avhen these two jiarts generate inpial Vi^himes, 

212. The moment of a force about an axis perjHMulic- 
ular to its line ot" direction is the product cd" its magnitutlo by 
the piM-piMulicular distance oi' its lim> of action irom the axis. 



226 



INTEGRAL CALCULUS. 



and measures the tendency of the force to produce rotation 
about the axis. 

The force exerted by gravity on a body varies as the mass 
of the body, and may be measured by the mass. 

The centre of mass of a body is a point so situated that the 
force of gravity produces no tendency in the body to rotate 
about any axis passing through this point. 

The mass of any homogeneous body is the product of its 
volume by its density. 




213. To find the centre of mass of a body. 
Let the points of the body be referred to the rectangular 

axes OX, OY, OZ, the plane XY 
being horizontal. Let m denote 
the mass of the body, and 31 the 
moment of the force of gravity on 
j^ j^ m about an axis parallel to OZ 
and passing through C, (x, y, z). 

Let P be any point {x, ?/, z) in 
the body, and Q the point 
(x + Ace, y + A?/, ;*; + As). 
Let Am equal the mass of the parallelopiped PQ) then 
(x — x) ^m <i AJ/< (^x -\- Ax — x) A?>^. 
Am = point P ; 
Ax = i, Am = Vgi^, 

AM = (x — x) Am + vi", where n > 3. 
. * . d3f = (x — x) dm ; 

.'. M = i x dm — X i dm. (1) 

When (x, y, z) is the centre of mass, M=0\ hence from (1) 
a:; = j xdiiij I diin. [1] 



Let 
then, if 
and 



CENTRE OF MASS. 227 

In like manner we obtain 

— _fy ^^^^^' - _ /^ ^^ 

^ ^ fdm ' " "" /dm ' L J 

To obtain z place the ?>axis horizontal. 

Whether the body is homogeneous or not, dm denotes the 
mass of a homogeneous solid whose density is that of the 
body at the point F (x, y, z), (§ 11). 

Hence; denoting the volume of dm by dv and the density of 
the body at P by k, we have dm = kdv. Substituting kdv for 
dm in [1] and [2], we have 

~ fkdv ^ ~ fkdv '^ ~ fkdv L^-l 

When the body is not homogeneous, k is some function of 
the coordinates of the point {x, y, z), and 

dv = 31./JS V = dx dydz. 

When the body is homogeneous, k is constant, and formulas 
[3] become 

fdv ^ fdv "- fdv L^J 

CoR. In formulas [4] dr may equal 3\,,J\ Q'V, or dV. 

For when the body is homogeneous, all points in the plane 
ARP have the same moment. Hence, to prove [4] we may 
let Am equal the mass of the body between the planes J A7' 
and XNQ, or an increment of this mass. In the first case dr 
will equal dV, and in the second it will ecpial 9^,,/' or 9-|,, T. 

214. Centre of mass of right cylinders and areas. Let 
c denote the altitude of [\w rii;ht cvliniU^r whose tuuivex sur- 
face is made up of the cvlindrieal surtaees // j'.w // i^.r, 
and the })la.nes .r --- .r„, .»• — a\, t\\o })hine A' )' beini;- midway 
between and })aralhd to tlie bases. 



228 



INTEGRAL CALCULUS. 



Evidently i = 0. 

To find X and y we have da = 9^^ V = cdx dy. 

Hence, from [4] of § 213, we have 

- ffxdxdy — ffydxdy 

JJdx dy ffdx dy 



[5] 



the limits for x being x^ and x-^, and for ?/, fx and <^a7. 

As the values of x and y depend solely on the plane area 
bounded by the plane curves y = fx, y — cf)X, and the lines 
X = Xf), X = Xi; for convenience the point (x, y) is called the 
mass-centre of this area. 

CoR. 1. If a plane area be revolved about a line through 
its mass-centre, tlie two parts will generate equal volumes. 

For let this line coincide with the cc-axis ; then y = 0, and 
from [5] we have 

y dx dy = 0. (1) 



'ff' 



Hence, by Cor. of § 211, the two volumes are equal. 

Cor. 2. If an area is symmetrical with respect to the 
ic-axis, y = 0, and x is the same for one of the symmetrical 
halves as for the whole area. 



215. Centre of mass of rods and curves. Suppose the 
mass-centre of the plane figure CD to move along the curve 

HPB, its plane being always 
perpendicular to the curve. 

Let A denote the constant area 
CD; V, the volume of the rod 
generated by CD ; s, the arc HP ; 
and As, PP'. Through P' draw 
in the plane CD" the line P'n 
parallel to the plane CD'. On 
P'n as an axis revolve CD" until it becomes parallel to CD'. 




CENTRE OF MASS. 229 

Then, by Cor. 1 of § 214, we have 

AV = A- As -{- vi^, where ?i > 1 . 
.-.dV^Ads. (1) 

Substituting Ads for dv in [4] of § 213, we obtain 

- fxds — fy ds - fzds ^ ^ _, 

As the values of x, y, and z depend solely on the curve HB, 
for convenience (x, y, i) is often called the mass-centre of the 
curve HB. 

If the curve is in the plane XY, ii = ; if in addition it is 
symmetrical with respect to the a?-axis, y = 0, and x is the 
same for one of the symmetrical halves as for the whole curve. 

CoR. From (1) we obtain V = As. (2) 

That is, the volume of tlie rod CDF equals the area of CD 
into the length of the arc traced by the mass-centre of CI). 

The proofs of ecjuations (1) and (2) fail when CD cuts the cvokite of the 
curve IIB. 

EXAMPLES. 

1. Find the centre of mass of the area hounded by the parabohx 
y'2 = 4px and a double ordinate. 

From the symmetry of the curve, y = 0, and 



x= i f '"''xdxdi// \ ) 



dx di/ = I) J' / 5. 



2. Find the centre of mass of the area bounded by (lie semicubical 
parabola ((//'- = .r> and a double ordinate. A us. x = ox /I. 

3. Find the centre of mass of the area bounded by the //-axis and the 
curve x>/^ =- l)^ {a - .v). ^^s. x = a/ 4. 

4. Find tlie centre of mass of the area o\' {\\c lirst tiuadrant o\' the 
ellipse x-/a' + ir/b- - 1. ^i ,j.s-. .r - -I a/:\ ,t, u - ^ ^/'> ^' 



230 



INTEGRAL CALCULUS. 



5. Find the centre of mass E of any circular arc BOB. 

D Let OB = OB = -s, being the origin. 

The equation of BOB is y- = 2rx — x^, r 
being the radius. 

From the symmetry of the curve 




and 



fo^xds _ r 
fods ~s 



Jo Vi 



xdx 



rx — xP- 
= r — ry / s — OE. 
Hence, CE = 2ry/2s = r chord BB/arc BOB. 

6. Find the centre of mass of the arc in the first quadrant of the curve 



x2/3 + y2- 



Ans. X = y = 2a/6. 



7. Find the centre of mass of the arc of a catenary cut off by any 
horizontal chord. 

Ans. y = {ax + ys) /2 s, where 2 s is the length of the arc. 

8. Find the centre of mass of the curve xy^ = b'^ {a — x). 

Ans. x = a/4:. 

9. The axis of a homogeneous solid of revolution is the x-axis ; show 
that V =1: = 0, and 



X = j j xydxdy/ i i V 



dxdy. 



211 



10. Find the volume of the ring generated by the revolution of an 
ellipse about an external axis in its own plane, the distance of the centre 
of the ellipse from the axis being r. 

Ans. 2 Tt^ahr. § 215, Cor. 



11. If an arc of a plane curve revolve through d radians about an 
external axis in its own plane, the area of the surface generated will be 
equal to the length of the revolving arc, multiplied by the length of the 
path described by the mass-centre of this arc. 
From [6] of § 215 we obtain 



By-s 



•/o 



yds., or the theorem. 



§203 



MOMENT OF INERTIA. 231 

12. rind the surface of the ring generated by the revolution of a circle 
(radius a) about an external axis, the distance of the centre of the circle 
from the axis being r. Ans. 4 tt^ ar. 



216. The moment of inertia of a plane area about a given 
point in its plane is the limit of the sum of the products 
obtained by multiplying the area of each infinitesimal portion 
by the square of its distance from the given point. 

Denote by M.I. the moment of inertia of the area NBCD, 
or A, about (§ 207, fig.). Let P be any point {x, y) in this 
area, x and y being independent ; then 

op' = x' + y\ 
Let PF = Ax = i, and PG = Ay = vl ; 

then /ll^^(j}LI.) = (x'^ + y^) Ax Ay + vi'', where /i. > 2. 

.■.Ql(M.L) = (x'-\-f)dxdy. 

.'.ILT.=ff(x^ + f)dxdy, 

the limits being taken so as to include the re(pured area. 

Ex. 1. Eind the moment of inertia about tlic origin, of the circle 
x'2 + y2 = a~. 



Ex. 2. Eiiul the nionuMit. of iiuM'tia about ihc oriuiu, o{ tlu- sinalK-r area 
bounded by the .c-axis, [hv parabola //'- ■ 1 ii.i\ and ilu> Hue .r + // = ;>(/. 

Jo Jg-/Ui "^^ 



CHAPTER IX. 



DEFINITE INTEGKAL AS A LIMIT. INTRINSIC EQUATIONS 
OF CURVES. 

217. Definite integral as a limit. Heretofore we have 
considered differentials as finite; in this article we shall 
regard them as infinitesimal. A definite integral has been 
defined as an increment of an indefinite integral. We pro- 
ceed to show that a definite integral equals the limit of the 
sum of an infinite number of infinitesimal differentials. 

j5 To make the theorem and its 

proof as clear as possible, let 
ns consider the area MiP^BX, 
which we will denote by A. 

Let Oilii = a, OX = h, and 
P^B be the locus oi y = cf>x. 
Divide II^X into 71 equal 







■ 






P. 


/] 


1 


. 1 


3 I 




Qz 







Qn 



O M, 3f, M, 



MnX 



parts, M^M^, M^M^, 
and divide the area A as in the fissure. 



MX 



Let 
then 



dx 



M,M, = 3I,3f, '. 
MiQi = (fi(a)dx, 
MgQs = (f>(a + 2 dx) dx, 
.'. A = (f)(a)dx H- (f>(a H 



• • • = M,,X, 
M2Q2 = <f>(a -\- dx) dx, 

• ', M^Q^ = 4>{b-dx)dx. 
dx) dx -\- (f>(a -h 2 dx) dx 

H -\-cf>(b-dx)dx-{- T, (1) 

where T is the sum of the triangles 

PiQiP^j PiQi^z-) ' ' ': PfiQu-^' 

By the notation of sums, (1) is written 



DEFINITE INTEGRAL AS A LIMIT. 233 

Evidently T<XB-dx, or (fihdx. 
Let 71 = ^'^ then t/a? = 0; z. T = 0. 

••• /'"!'' x''*(*)^^* = ^ = rV(-*)'&- (2) 

The first member of (2) denotes the limit of the sum of an 
infinite number of differentials, each of which is rej^resented 
by ^x dx, X taking in succession the values 

a, a -\- dx^ a H- 2 dx, • - -, b — dx, while dx =^0. 

When (jix is constant, T = 0, and A equals the sum of the 
differentials in (1) for all values of dx. 

Again consider the volume generated by revolving M^P^BX 
about OX as an axis. Each of the rectangles M^Q^^, ^UQ^, ' ' ', 
MjiQn will generate a cylinder whose volume will be repre- 
sented by TT (</).^')^ dx. Hence, 

V = ^'''7r(cf,xydx + T, where T< '7r(<f>hy'dx. 



limit 
dx 



''''\ X'^ (<^.^')'^^^ =V= f'TT (cf^xf dx. § 203 



EXAMPLES. 



1. The effect of gravity in inakiiig a body tend to rotate about any 
given axis is the same as if its mass were concentrated at its centre of 
mass. 

From [1] of § 21o, by integration Ave liavc 

- /•'" , limit x-^'" ■, 

xm= I xd)ti= , ,^ > xdin. (\) 

Jo '^"' = ^^ ^.. ^ ' 

The given axis being OY, xin is what wouhl be the moment of tlio 
force of gravity on vi if vi Avere concentrateil at its centre of m;is.^. 
The last member of (I) is tl\i> limit of tlu> sum o{ the monuMits of the 
force of gravity on all tlu> maliM-ial (Huuts [(Ini) of in when (///; £r (\ 
Hence, (1) proves the tlu-oroui. 

2. Show that tlu> area of a polar mirvc is ilic liu\ii oi the sum i^f an 
intinit(> niunbiM- oi inthiitc'simal iliffcrcntials. 



234 INTEGRAL CALCULUS. 

3. Using the figure in § 207 sliow that 

> dxdy = {<px — fx) dx= \ dx dy, (1) 

'^fx Jfx 

limit ^-^1 r^\ 

and dx — ^ (^^ — fx)dx — A= I (0x — fx) dx. (2) 

(1) holds true whether dy is finite or infinitesimal ; for dx being 
constant the sum is constant. 

In (2) the sum varies with dx, and dx must be infinitesimal to 
cause this sum to approach its limit A. 

4. Using the figure in § 210, show that 

dxdydz = NRdxdy = I dxdydz; (1) 

•/o 



limit -O^^^ r^^^^ 

A 2j NRdxdy = AMB-dx= | NRdxdy; (2) 

= -^^0 «yo 



dy 

hmit ^^ox fox 

^Zj AMBdx = OZY-X = | AMB ■ dx. (3) 

In (1) dx and d?/ are constants, and dz may be either a constant or 
an infinitesimal. In (2) dx is a constant, but dy = 0. 



218. Intrinsic equation of a curve. Let s denote the arc 
between a fixed point, Q, and a variable point, P, of the curve 
QP, and r the angle ABF included between 
the tangents at Q and P ; then the equation 
which expresses the relation betw^een the 
variables s and r is called the iiitrinsic equa- 
tion of the curve. 

j^ Ex. 1. Eind the intrinsic equation of the circle. 

Let QP, or s, be an arc of a circle whose radius is r. 
Let C denote the centre of this circle ; then 
r = Z ^BP = Z QCP = s/r. 

Hence, s = rr is the intrinsic equation of the circle. 




INTRINSIC EQUATIONS. 



235 



Ex. 2. Find the intrinsic equation of the catenary. 

In § 197 let OB = s; then r = Z XAB ; 
and tan r = dy/dx = s/a. 

Hence, s = a tan r 

is the intrinsic equation of the catenary. 

Ex. 3. Find the intrinsic equation of the tractrix. 
In § 198 let ^P = s ; then sec t = secEPT= a/y. 
Hence, 8 — a log sec r 

is the intrinsic equation of the tractrix. 



197, (1) 



§ 198, (2) 



219. To obtain the intrinsic equation of a curve from its 
rectangular or polar equation, we find the values of s and r 
and eliminate the other variables between these equations. 

Ex. Find the intrinsic equation of the cycloid. 

When s is reckoned from the cusp (§ 196, example 2), we have 



s = 4 r (1 — V2 r — y / v2r) 
and cos r = dy /ds = V2 r — y/ V2 r, 

.-. s = 4r{l — cost). 
When s is reckoned from the vertex, we have 

s= —^2r 



(1) 



^2 r j ' (2 r - y)- 1 ^--^ d?/ = 4 r • V2 r — y / \'2 r 



and 



sin T — — dy /ds 
.-. H — 4r sin T. 



V2r 



y/V2 



(2) 



220. If the inf)-uisic ('(/utftio/i of the iin'olute QP is s = fr, 
tJie iiUrinsic eqiaxtloii of ihc erolute Qil\ is 

s=.fV-f'0. (1) 

The curvature of ()/* is dr /ds. 

.'. E = ds /(It =f'T. 

= f'r-f'0 
= /'ti-/'0, 
since ti = t. 

Oinittini;- tlu> subscripts in (^,'>V wo have (1). 




236 INTEGKAL CALCULUS. 



Ex. 1. The evolute of the tractrix s = a log sec r is 



d (log sec t)' 
dr 



s = a " '"'° = a tan r, 

Ju 



which, by example 2 in § 218, is the catenary. 

Ex. 2. The evolute of the cycloid s = 4 r (1 — cos t) is 

. d{l— cos t)-| ^ ^ . 

s = ir -^ ; = 4 r sm r, 

dr Jo 

which, by the example in § 219, is an equal cycloid. 



EXAMPLES. 

1. Find the evolute of the catenary s = a tan r. 

2. Find the intrinsic equation of x^^^ + y-^^ =^ a-^^ and of its evolute. 

Ans. s = (3 a/ 2) sin- r ; s = (3 a/ 2) sin2 t. 

3. Find the intrinsic equation of the logarithmic spiral p = 6e^/« and 
of its evolute. 

When s is measured from the point (6, 0) where the spiral crosses 
the initial line, we have 



s = 6V1 + a- {eo/a — 1). § 199, example 2 
Since xp is constant, t = 6. 



... s = bVl +a^(e^/«-l). 



CHAPTEE X. 

ORDINARY DIFFERENTIAL EQUATIONS. 

221. A differential equation is an equation which involves 
one or more differentials or derivatives. 

An ordinary differential equation is one which involves only 
one independent variable. 

For example, dy = cofixdx^ (1) 

d'^rj/dx^ + y = 0, (2) 

and y = x-dy/dx-\- r^l + {dij/dx)-, (3) 

are ordinary differential equations. 

The order of a differential equation is the order of the 
highest differential or derivative which it contains. 

The degree of a differential equation is that of the highest 
power to which the highest differential or derivative wliieh it 
contains is raised, after the equation is freed from fractions 
and radicals. 

Thus equation (1) is of the first order and first degree, (2) is of the 
second order and first degree, while (8) is of the first order and second 
degree. 

222. The general solution of a diffiM-ential e(]uation is the 
most general equation free from dirferentials or derivatives. 
from which the former equation may be dtu-ived by differen- 
tiation. 

The general solution of (Hjuation (1) in § 221 is 
// -- sin X + (\ 
whei-e (1 is the roufitu))! of i)itciini(i(^)i. 

y = sin .r, // = sin.r + 7, • • •. arc partirnlar solntions of (1), whioh are 
included in its ucMii^ral solntion. 



238 INTEGRAL CALCULUS. 

The general solution may not include all possible solutions. A solu- 
tion not included in the general solution is called a singular solution. 

For a discussion of singular solutions the reader will consult some 
treatise on differential equations. 

The general solution of a differential equation of the nth 
order contains n arbitrary constants of integration. It is often 
called the complete integral or primitive of the differential 
equation. 

223. In the foregoing chapters it was our object to obtain 
the general solution of differential equations of the form 

dy = (f) (x) dx. 

In this chapter we shall extend the process of integration 
to differential equations of the more general form 

Mdx 4- Ndy = 0, (1) 

where Ji"and i\^are functions of x and y. 

The variables in Mdx + Ndy are said to be separated when 
-M, or the coefficient of dx., contains x only, and N contains 
y only. 

When Mdx + Ndy is the total differential of some function 
of X and ?/, it is called an exact differential, and (1) is called 
an exact differential equation. 

For example, xdy + ijdx is the exact differential of xy ; hence, the 
general solution of the exact differential equation, 

xdy -{- ydx = 0, 
is xy = C. 

224. Equations of the form (^i (x) dx + ^2 (y) dy = 0. (1) 

When, as in (1), the variables are separated, a differential 
equation is solved by integrating its terms separately. 
For example, the general solution of the differential equation 

e^dx + Sij-dy — 0, 
is e* + y3 = (7. • 



VARIABLES SEPARATED. 239 

When an equation is, or may be, written in the form 
<^i (^) • </>2 {y) dx + <^3 (x) ■ <^4 (ij) dy = 0, 
the variables may be separated by dividing both members by 
^2(u) •</>3(^). 

Ex. 1. Solve (1 - x) dy - (1 + y) dx = 0. (1) 

Dividing by (1 — x) (1 + y) to separate the variables, and integrating, 
we obtain 

log (1 + 2/\ + log (1 - X) = log C. (2) 

.■.{l+y){l-x) = C. (3) 

Equations (2) and (3) are two equally correct ways of expressing the 
general solution of (1). 

Solution (3) could be obtained without separating the variables in 
(1) by noting that (1 — x) dy — {I -\- y) dx is the exact differential of 
(l-x)(l+2/). 

Ex. 2. Solve (x2 + 1) d^J = (7/2 + 1) dx. 

x -\- c 



Here tan- 1 y = tan- ^ x + tan-^ c = tan— ^ 

.•• y 



\ — ex 

X + c _ 
1 — ex 



EXAMPLES. 
Solve each of the following differential equations : 

1 "k. ^ ^'l±JL±.l . 3;^ - ?/' , '^'- - U~ _L , _ „ ^ r 

' dx y^ + y+l 3 "^ 2 "^ ^ 

^ + logx = ^ + ^+C. 

log (.(•//) + .r - 1/ = C. 

',] ((•" - e>) rr .r i + C. 

5. X cos'- // dx = y cos"- xdy. 

X tan .)• — log si>c .c " - // tan // — log sec // + ('. 

6. a {xdy -\- 2 ydx) ~ xy dy. x'-y — €€»'». 

7. Find tlu> ciiuation of the t'aniilv of cuvvi's' whosr slope is 

(t .L- + 2 .r + 1) / ((; //•- + J //). 2 //^> -f 2 //••i ~ .r-» + .r- + .r + T. 



2. 


X2 + 1 dy 
y^X='Ux 




3. 


(1 +x)ydx + (1 -y)xdy-- 


= 0. 


4. 


dy = (e^-!i + x"-c-") dx. 





240 INTEGRAL CALCULUS. 

8. The equation of the family of curves which cross all their radii 

vectores at the same angle ^ is p = Ce^^^o*^. 

Here dp/p = cot Add ; .-. log p = loge^^otA -\. \Qg q^ 

9. Find the equation of the family of curves 

(1) whose slope is — Ifix/a^y ; 

(2) whose slope is (e-'^/" — e-^/«)/2 ; 

(3) whose subtangent is the constant a ; 

(4) whose subnormal is the constant a ; 

(5) whose tangent is the constant a. 

Ans. ahj'^ + 6-2x2 =: C ; ?/ = a (e^/« + e-^/«) /2 + C ; y= Ce^'^ ; 

?/2 = 2 ax + C ; x — ^ d?- — y- + a log [(a — Va^ _ y-i^^ jy-^ + q^ 

10. Find the equation of the curve which passes through the point 
(a, 6), and intersects at right angles each of the series of curves repre- 
sented by the equation 

?/2 = 2 ax^, 
in which a: is a variable parameter. 

11. Helmholtz's equation for the strength of an electric current, C, at 

Xp 1(10 

the* time ^ is 0= — — — -77' where E^ R, and L are given constants. 
i\ K dt 

Find the value of C, having given that C = when t = Q. 

Ans. C = E{l-e-Jii'r.)/ji. 

225. Equations homogeneous in x and y. After being 
divided by x"- (11 being the degree of each term in x and y), 
any equation homogeneous in x and y can be put in the form 

dy=f{y/x)dx. (1) 

Putting y = vx in (1), we obtain 

V dx + x dv = f(v) dx. (2) 

The variables in (2) are easily separated ; hence, its solu- 
tion is found by § 224. 

Ex. L Solve (x2 + y'i)dx = 2xydy. . (1) 

Putting y = vx and dividing by x^, we obtain 

{l+v'^)dx = 2v{xdv-i- vdx). (2) 



HOMOGENEOUS EQUATIONS. 241 

Separating the variables and integrating, we have 

log[x(l -u2)] = logC. 
Putting y /x for v, the solution becomes 

^2 _ ^2 = Qx. 

Ex. 2. Solve (x2 + ?/2) (Z?/ = xydx. (1) 

Putting ?/ = I3X and dividing by x^, v^e obtain 

(1 + xP) (x dv + D dx) = V dx. (2) 

Separating the variables and integrating, we have 
logx + log C = u2/2 — logu, 
or C?/ = e^^/2.v2^ 

EXAMPLES. 
Solve each of the differential equations : 

1. x'^dy — y-dx = xy dx. log x + x/y = C. 

2. (2 v^ - X) dy ■^ydx = 0. y= Ce~^^. 

3. yHx + x:M.y = xydy. y = Cey/-^. 



4. xdy = {y + V^--^ + y-^) dx. ' x^ = C"-^ + 2 Cy. 

5. (x + ?/) (/// = (y — x) fZx. log (x- + ?/-) + 2 tan-i (y/x) == C. 

6. x'kly = y-dx. y — x= Cxy. 

1. (8 ?/ + 10 X) dx + (5 y + 7 x) (/// = 0. (// + .)•)•-'(// + 2 x)" = C. 

8. Find the system of curves at any point of which, as (x, y), the sub- 
tangent is equal to the sum of x and y. An&. y ~ Co^'f". 

226. Non-homogeneous equations of the first degree in 
X and y. 

These equations are of tlie form 

{a.v + hi/ + c) ilx -h {a\r + //// + r') il ,/ = 0. ^H 

Puttino- J'' -f- Ji I'oi- X and //' + /,• for // in {\^, wi^ obtain 
{a.v^^h,/' + ah +/V/ + rV/.r 

-f {^<i'x' + ////' -h n'h -f /''A- -h c'^ (/// -- 0. (2) 



242 INTEGRAL CALCULUS. 

Giving to h and k the values determined by the system, 

ah + hk-{-c = 0, a'h + h'k -\-c' = 0, (3) 

equation (2) becomes 

{ax' + hi/') dx + {a'x' + b'lj') dy = 0, (4) 

which is homogeneous in x' and y', and can therefore be solved 
by the method of § 225. 

This method fails when a' /a = b' /h ; for then h and k in 
system (3) are infinite or indeterminate. 

In this case assume 

a' /a = b' /b = m, or a' = ma, b' = mb. 

Equation (1) then assumes the form 

(ax -\- by + c) dx + [^m (ax + by) + c'] dy = 0. (5) 

Let ax -^ by = v; then dy = (dv — adx) /b. 
Substituting these values in (5), we obtain 

\b (y -\- c)— a {rriv + c')] dx + {mv + c') dv = 0, 

where the variables are readily separated. 

EXAMPLES. 
1. Solve (2cc + 32/-8)(?a;-(a: + ?/-3)£Z?/ = 0. (1) 

Putting x' -\r h for x, and y' •{- k for ?/, we obtain 
(2 X' + 3 ?/' + 2 ^ + 3 A; - 8) dx' = (X' + ?/' + /i + A; - 3) tZ?/'. (2) 

Assume the system 

2^ + 3A:-8 = 0, /i + A:-3 = 0; or 7^ = l, A; = 2. 
Equation (2) then becomes 

(2 X' + 3 y') dx' = {x' + y') dy\ (3) 

Putting ?/' = vx', (3) becomes 

(2 + 3 v) dx' = {l+v) (vdx' + x'dv), 



NON-HOMOGENEOUS EQUATIONS. 243 

dx' V + 1 -, 

x' {v — ly^ — 3 

" L(^ - 1)'^ - 3 "^ Vl (^ - 1 - V3 ~ « - 1 + V?)\ '^''' 
.-. - logx' = \ \og{(v - 1)2 - 3} + ^log ^~|7^ + C. 

^ V 3 V — 1 + Va 

where x' — x — \ and v = '- • 

X — 1 

2. (3 ?/ - 7 X + 7) dx + (7 ?/ - 3 X + 3) dy = 0. 

J.ns. (?/ - X + 1)2 {y + x- 1)5 = C. 

3. (2 X + ?/ + 1) dx + (4 X + 2 y - 1) dy = 0. 

Ans. X + 2y + \og{2x -\- y — 1)= C. 

4. (2 ?/ + X + 1) cfx = (2 X + 4 ?/ + 3) dy. 

Ans. 4x — 8 ?/ = log (4x + 8 ?/ -I- 5) + C. 

5. (7 ?/ + X + 2) dx = (3 X + 5 ?/ + 6) (Z?/. 

^7is. x + 5?/+2 = C(x — ?/ + 2)-*. 

227. Exact differential equations. The condition that 

Mdx + Ndy - (1) 

m,ay he an exact differential eqiiation is 

aM/dy = QN/dx. (2) 

Comparing (1) with tlio exact dilTerential tMinatic^i 

du = ^"dx + ^-d>/ = 0, (3) 

ax at/ ' ^ ^ 

wc obtain 

]\r=3u/dx, X=3u/df/, (4) 

as the conditions that (1) ho r.vacf. 

From conditions (1) by (HlTiMnMitiniion. \V(^ obtain 

3M ^ Q^H ^3N^ ^^^. .>. 

(/// d//d.v dx ' 



244 INTEGRAL CALCULUS. 

Condition (2) is called Euler's Criterion of Integrability. 
When condition (2) is satisfied, (1) may be solved by regard- 
ing y as constant and putting 

u= \ Mdx -\-fy, (5) 

and then determining/^ so that 

'du/dy = N. ' (6) 

Or regarding x as constant, we may put 

u = JNdy+fx, 

and so determine fx that 

Qu/dx = M. 
Equations (5) and (6) involve the conditions in (4). 

Ex. Solve X (X + 2 y) dx + {x^ - y^) dy = 0. (1) 

Here M=x{x + 2y), N = x^ - y^. (2) 

.-. dM/dy = 2x = dN/dx ; 

hence, condition (2) of § 227 is fulfilled. 
Regarding y as constant, we put 

u= i x{x-\- 2y)dx+fy 

= xyS + 7jx'-+fy. (3) 

To determine fy, from (3) and (2) we have 

du/dy = x2 +f'y = N = x- — ifi, 
.'. ry = - y\ or fy = - y^S. (4) 

Erom (3) and (4) we obtain 

w = xV3 + ?/x2-?/V3. 
.-. x3 + 3 ?/x2 -y^= C 
is a solution of (1). 



INTEGRATING FACTOR. 245 

EXAMPLES. 
Solve each of the following differential equations : 

1. (6 xy - ?/2) dx + (3 x2 - 2 xy) dy = 0. 3 x'^y - y^x = C. 

2. {x^ + 3 x?/) dx + {if + 3 xhj) dy = 0. x^ + 6 x'V + y^ = C. 

3. (X2 + ?/2) C?X + 2 XT/ d?/ = 0. x3 + 3 x?/2 = C. 

4. (x2 — 4Lxy — 2y-)dx + (ij- — 4xy — 2 x^) dy = 0. 

x3 — 6 x2?/ — 6 x?/2 + 77^ = C. 

5. (1 + ?/2/x2)dx - (2 ?//x)cZ?/ = 0. x^-y^= Cx. 

6. X dx + ?/ dy -\ , , -^^ =0. x2 + ?/2 + 2 tan-i - = C. 

x-^ + y^ X 

In example 6 divide both terms of the fraction by x2. 

7. e^ (x2 + ?/2 + 2 x) dx + 2 ye^ d?/ = 0. e^ (x2 + ?/2) = C. 

8. (2 ax + 6?/ + (;) dx + (2 c?/ + 6x + e) cZ?/ = 0. 

228. Integrating factor. When the equation, 
Mdx 4- Ndy = 0, 
is not exact, it may sometimes be made exact by multiplying 
it by a factor called an integrating factor. 

Sometimes an integrating factor may be found by inspec- 
tion, as in tlie examples below : 

Ex. 1. Solve ydx — xdy = 0. (1) 

Equation (1) is not exact, but when multiplied by //--', it becomes 

ydx — xdy _ 
y-i - ' 

which is exact, and which has for its solution 

x/y = 0. (2) 

Multiplied by 1 /•*'//, (1) becomes 

dx /x — dy / y = 0, 
which is exact, and has for its solution 

loj;- (.)•///)= log C. (S) 

Solution (-J) is readily obtained from (;>). 
INIulliplying (1) by .(—-, we obtain thr solution 

y/x = i\. 



246 INTEGRAL CALCULUS. 

Ex. 2. Solve (1 + xtj) ydx + (1 — xy)xdij = 0. (1) 

From (1), ydx + xdy + xy'^dx — dS^ydy — 0, 

or d (xy) + xy'^dx — xHjdy — 0. (2) 

Dividing (2) by x2?/2, we obtain 

(x?/)2 X y 

1 X 

.'. 1- log - = log C, or X = Cye^i^J. 

xy "^y "= ' 

229. Rules for finding the integrating factor of 

Mdx + Ndy = o. (a) 

We give below the rules in four cases : 
EuLE I. When Mx + Ny is not equal to zero, and (a) is 
homogeneous, {Ilx + Nyy^ is an integrating factor of (a). 

Ex. Solve {xHj — 2 xy"^) dx — (x^ — 3 xhj) dy = 0. (1) 

Equation (1) is homogeneous, and 

Mx-\-Ny = x^y — 2 x'^y^ — xHj + 3 x2?/2 = x'^y\ 
Hence, (x2?/2)— i is an integrating factor of (1). 
Dividing (1) by x^?/^, we obtain the exact equation 

Solving (2) by § 227, we obtain 

x/?/ + log(7/Vx2)= C. 

EuLE II. When Mx — Ny is not equal to zero, and (a) is 
of the form 

/i {^y) y dx + /a {xij) X dy = 0, (b) 

(Mx — Ny)~'^ is an integrating factor of (a). 

Ex. Solve (x22/2 + xy) ydx + (x2?/2 — l)xdy-0. (1) 

Equation (1) is of the form of (b), and 

Mx — Ny = x2?/2 + xy. 
Hence, {x'^y^ + xy)—'^ is an integrating factor of (1). 
Divided by x2?/2 + xy, (1) becomes 

ydx + xdy = dy/y. 
.: xy + log C = log y, or y = Ce^. 



INTEGRATING EACTOR. 247 

Rule III. When —- f — -— ) is a function of x alone, 

NKdydxJ ' 

say/c, g//(a;)(?a; jg 2ji integrating factor of (a). 

Ex. Solve (x2 + 2/2 + 2 x) dx + 2 ?/ dy - 0. (1) 

Hence, e-/>"(^)^'^ = e/''^^ = e^, the integrating factor of (1). 
Multiplying (1) by e^, we obtain the exact equation 
e^ (x2 + ?/2 + 2 x) dx + 2 e^?/ dy = 0. 

.-. e^ (x2 + ?/2) = C. § 227, example 7 

Rule IV. When -- ( — ) is a function of ?/ alone, 

M \dx dy J J ^ 

^^J fl/f eX^'(z/)'^y is an integrating factor of (a). 

230. Proof of rules in § 229. I. When Mdx + ^^dy is 
homogeneous, 

3Idx + Ndy = \ \{Mx + Ny) (— H- ^ j 

+ (J/.r-.A7/)./log(.r///)]. (H 

Jfr/x- + Nd,y 1 , , , , , IJfa^ - i\^// , , .r 
Mx + Ny 2 ^^ ^'^ 1> J/;,. _|- :\ ^ ^~^ ^ 

^2^nog(.n/) + ^./y-i;-^- (2) 

7l/:r - Ni/ . ., ^ , . . . , , 

T— — ; — — r- IS evulentlv cMuial to sonu^ t unction ot .r u. \\\w\\ 
Mr -\- Ny ' ^ 

TIf and iVare honioircnoous. 



248 INTEGRAL CALCULUS. 

The second member of (2) is an exact differential ; hence, 
{]\Ix -\- Ny)~^ is an integrating factor of (a). 

Equation (2) fails when Mx -\- Ny = 0. But in this case 

M/N=-y/x. 

Substituting this value of M/ N in (a) of § 229, we obtain 

log {y/x) = log C,'OV y = Cx. 

II. When Mdx + ^dy is of the form 

/i (^^) ydx-\-f^ {xy) X dy. 
Dividing (1) by Mx — Ny, we obtain 
Mdx + Ndy 1 Mx + Ny x _l 1 7 A A /Q^ 

—rz TT^ = - ITT T7^ d (log Xl/) + 7: fZ lt>S - (3) 

Mx — Ny 2Mx- Ny ^ ^ '^^ 2 \ ^ yj ^ ^ 



2f^(xy)xy-f^{xy)xy xy 2 \ y 

The second member of (4) is an exact differential ; hence, 
{Mx — Ny)~'^ is an integrating factor of (a). 

III. When-^— -— j=/.. 

Multiplying (a) by g/^^^)^'^, we obtain 

eff(-)^i^Mdx + e-^-^^'^^'^^Ndy = 0, 
which is exact, for by differentiation we find that 



dy^ ^ dx^ ^ 

In like manner Eule IV is proved. 



227 



LINEAR EQUATIONS. 249 

EXAMPLES. 

1. (x2 + 2xy- ?/2) dx = {x'^-2xi/- if-) dy. 

2. {X"tf + xy^) dx = {xhj + xhf) dy. 

X y \y X / 

4. xHx + (3 x'^y + 2 y-^) dy - 0. 

5. (1 + xy) ydx + {I— xy) x dy — 0, 

6. {Vxy — l)xdy = {Vxy -{- l)ydx. 
'''• {y + y^^'xy)dx + [x + x'yx^j)dy = 0. 

8, e^-^ (x^z/^ + x?/) (x(^?/ + ydx)-\- ydx — xdy = 0. x?/e-^'/ = log (Cy/x). 

9. (3 x2 - ?/2) dy = 2 x?/ (^x. x2 - ?/2 = Cz/^. 
10. 2 x?/d?/ = (x2 + y2) cZx. ?/2 - X- = Cx. 

231. Linear equations of the first order. A Ihicfir differ- 
ential equation is one in which the dependent variable and 
its diffeientials appear only in the first degree. 

The foan of the linear equation of the first order is 





xM 


-2/2= C(x + 2/), 




y = 


Cx. 




X2 - 

+ 2y^ 


- Z/- + xy = C. 


x2 


= C^x^^+y\ 




X = 


Cye-!'. 


2/ 


Wxy = 


--\og{Cx/y). 




xy - 


= C. 



(hj + Fu<Li' = Qdr, (1) 

where F aid Q are functions of .r or are constants. 

The solutton of </// + Pi/d.v = 
is \ log // + log e-''^^'''- = log Cy 

or \ yc'^''''-^- = a (2) 

DillVrentiatin^ (2), w(> obtain 

whicli shows that //'''•'• is :ui integrating factor oi' (^H. 



250 INTEGRAL CALCULUS. 

Multiplying (1) by e-^^^^ and integrating the result, we obtain 

ye fPdx ^ r g fPdx Q ^^^ ^4^ 

Equality (4) may be used as a formula for solving any 
linear equation in the general form (1). 

EXAMPLES. 

1. {1 -h x^)d7/ — yxdx = adx. (1) 
Putting (1) in the general form, we obtain 

Hence, Cpdx = - C^^i = ^^S (^ + ^^)~^^^- 
... e/Pf7x = eiog(i + x2)-i/2 - (1 + x2)-i/2, 
and Je/P-(3.x = fj^f$p:. = j^:^, + C. 

Substituting tliese values in formula (4), we obtain 
y = ax + cVl + x2. 

c^ dif , ^ X 1 , „ „ 

2. x-r — mj = x + l. 7j = \ Cx«. 

dx 1 — a a 

3. dy + ydx= e-^dx. y = {x + C) er^. 



4. X (1 — X") dy + (2 x2 — 1) ?/ (Zx = ax^ dx. y = ax+ Cx^^l — x^. 

5. cos X • c??/ + ?/ sin X • (ix = dx. y = shix +0 cos x. 

6. (x2+ l)cZ?/ + 2x?/cZx = 4x2dx. 3(x2 + 1) ? = 4x3 + C. 

7. (Z?/ + 2/ cosx- dx= (l/2)sin 2x- dx. ?/ = sinx- 1 + Ce-^inx 

232. Equations reducible to the linear orm. Of such 
equations the most important are those of tie form 

dtj/dx+F7/=Qij-, (1) 

where F and Q are functions of x or consants. 



LINEAll EQUATIONS. 251 



i-w . 



Assume ^ — y 

1 n 

then y = z^-"", y" = z^-"". 



^ \ — n 

Substituting these values in (1), we obtain 
dzldx + (1 - 7^) P;^ = (1 - n) Q, 
which is linear in z. 



EXAMPLES. 

1. | + i. = xy. (1) 

Assume z — y—^\ 

then 2/ = 2;-l'^ dij- — {\lb)z-^i^dz, if-z-^'^. 

Substituting these values in (1), we obtain 

d^/dx — 5 2/x= — 5x2. (2) 

Solving (2) by § 231 and putting ?/— 5 for z, we have 
7/-5= (7x5 + 5xV2. 

2. (1 — x2) d5?/ = (ctxy- + xy) dx. y = (c Vl — x- — a)—^. 

x+ 1 1 



3. 3 yHy = (x + 1 + ay^) dx. y'^ = Ce"-^ - 



a 



4. (??/• ■■■= {x^y-^ — xy) dx. y- - = x- + 1 + Co'". 

5. ^ + ~y = 3 x-V^ ^=^. 7y-i/-= Cx- /^ - 3 X'". 
dx X 

6. (1 - X-) dy + xy dx = x//i / - dx. i;i/-2= C(\ - .f-) i ^ ■• + 1 . 

233. Equations of the first order and nth degree. Wo 
shall consider only such equations of tlio first c>riU>r ami ii{\\ 
degree as can be resolved into // equivalent rational equations 
of the first degree and of such types as have been sol veil in 
this chapter. 

The method will be made clear by the following example, 
where p = dy /dx. 



252 INTEGEAL CALCULUS. 

Ex. Solve p3 + 2 xp2 — ^/SpS _ 2 xif-^ = 0. (1) 

Equation (1) is equivalent to the equation 

p{p-{-2x){p- if-) = 0, 
which is equivalent to the three equations 

p = 0, i? + 2 X = 0, p — ?/ = 0. (2) 

Solving each of the equations (2), we obtain 

y =C, y + x2 = C, xy + Cy + \= 0, (3) 

where we have regarded all the constants of integration as equal. 
Combining the three equations in (3) into one, we obtain 

{y -C){y + x2 - C) {xy + C?/ + 1) = 0. (4) 

Equation (4), or the three equations in (3), is the solution of (1). 

EXAMPLES. 

1. i)2 _ 7p + 12 =: 0. (?/ - 4 X - C) (y - 3 x - C) = 0. 

2. p2 _ ax^ = 0. 25 {y + C)2 = 4 ax^. 

3. p2 _ 5p + e = 0. (?/ - 2 x - C) (?/ - 3x - C) = 0. 

4. J93 (a- + 2 ?/) + 3i)2 (^ + ^^) 4. (y + 2 x)p = 0. 

(2/ - C) (X + 2/ - C) {xy + X-+ y- - C) = 0. 

5. 4 ?/2p2 + 2px?/ (3 X + 1) + 3 x3 = 0. 

(X2 + 2 ?/2 - C) (X? + 2/2 -0=0. 

234. Equations of orders above the first. We shall illus- 
trate by examples the method of solving four special forms of 
such equations. 

I. Equations of the form d"y/dx" = fx. (a) 

Ex. Solve #?//cZx3 = 5 6x2. (1) 

Multiplying (1) by dx and integrating, we obtain 

dhj / dx2 = (5 / 3) 6x3 + d. (2) 

Multiplying (2) by dx and integrating, we obtain 

dy/dx = (5/12) 6x4 + CiX + C^. (3) 

.-. ?/ = 6x5/12 + Cix2/2 + C2X + C3. 



ORDER ABOVE THE FIRST. 253 

II. Equations of the form d^y/dx^ = fy. (b) 

Ex. Solve dHj/dx'^ + a^y = 0. (1) 

Multiplying (1) by 2d?/, we obtain 

.: {dy/dxY = - a'-y'- +Cr = a^ {c,^ - ?/2), (2) 

where Ci = a^c{^. 

From (2), dy /^c{^ — y- = adz. 

.-. sin— 1 (y /ci) = ax + C2, 
or y = ci sin (ax + C2). 

/d"v dv \ 

III. Equations of the form f [ j-~i • • •, ^, x j = o; (c) 

that is, equations of the nth order not containing // directly. 

Put p = -7-\ then -^ = ^, • • • — ^ = _\ ' 
dx dx^ dx c/x" dx" ^ 

Substituting these values in (c), we obtain 

which is an equation of the {11 — l)tli order between p and x. 

Ex. Solve dhj/dx- = d^ + h~ {dy/dx)'\ (1) 

Putting p = dy /dx, (1) becomes 

dp/dx = a- + b'p-. 
.-. tan-i {!)})/(() = ah{x+ C,), 
or bp = a tan \(d){x + Ti)]. 

.-. ?>-d// = tail [(tb{x + (\)] t(bdx. 
... Jy^i/ = log ivm" [ab{x + (^) j + (^. 

/d"v dv \ 

IV. Equations of the form f ( , ,/ ' ' '' a^' ^ ] ^' ^^^^ 






c^.f' '"'" </..^ '',//./.,= -''.v 



254 INTEGEAL CALCULUS. 

Substituting these values in (d), we obtain an equation of 
the (n — l)th order between j) and y. 

Ex. Solve d}^y/dx^ + a{dy/dx)-^ = 0. (1) 

Putting p = dy/dx, (1) becomes 

-T- + av — 0, or - = — ady. 

dy ^ ' p 

.•.p= Cier- «2/, or e'^i' dy = Ci dx. 

... ea?/ = Ci ax + C2. 

EXAMPLES. 
Solve each of the followmg equations : 

1. dhj = xe^dx^. y - x&- — 3 e^ + CiX^ + C^x + C'3. 

2. d4?y = a:3dx4. 

3. X d^y — 2 dx^. y — x'^ log x + CiX'^ + C2X + C3, 

where ci= Ci/2 -.3/2. 



4. ^2?/ = o?-y cZx2. ax = log (y + v ?/2 + ci) + C2, 

where a^ci = C\. 

5. ?/3 d^?/ = a dx^. Ciy^ = c^ (x + Co)^ + a. 

6. -s/mj d^y = dx^. 3x = 2 ai/4 (^^1/2 _ 2 ci) (?/i/2 + c^)i/2 + ^3. 

7. xd^y /dX" -\- dy/dx = 0. ?/ = Ci log x + C2. 

8. a2 {dhj/dx'^ )2 = 1+ {dy/dxY. 

2a-hj = cx&^'"- + Ci-^e-'^^« + C2. 

9. (1 + x2) . d^y/dx^ + 1 4- {dy/dxY = 0. 

y = Cix + (cr + 1) log (ci — x) + C2. 

10. (1 — x2) •ds^y/dxs — x-cZ?//dx = 2. 

y = Cisin— ix + (sin-ix)2 + c^. 

11. y -d^y/dx^ + {dy/dx)- = 1. ij'^ = x- + CiX + Co. 

12. The acceleration of a body moving toward a centre of attraction, 
C, varies directly as its distance from that centre ; determine the velocity 
and the time. 



APPLICATIONS TO MECHANICS. 255 

Let a = the acceleration at a unit's distance from C ; 

X = the varying, and c the initial, distance of the body from C; 
then xa = the acceleration at the distance x. 

Here s = c — x ; .-. v = ds/dt = — dx/dt ; (1) 

.-. xa = d^s/dt^ - — dH/dP. (2) 

Since v = — dx/dt = when x = c, by integrating (2) we have 
{dx/dty - ac2 - ax2. 



.-. V = — dx/dt - Va (c2 - x2). (3) 

Since i = when x — c, from (3) we have 

i = a-i/2cos-i(x/c). (4) 

Putting X = in (3) and (4), we obtain 

V = cVa, the velocity at the centre of force, C 
and i = (l/2)7ra-i/2^ (3/2)7ra-i/2, (5/2);ra-i^2^ • • •. (5) 

Hence the motion is periodic, the time-period being it /Va, which is 
entirely independent of the initial distance. 

The acceleration due to gravity at the earth's surface is 32.17 feet 
per second, and below the surface it varies as the distance from the 
centre. Hence, a particular case of the periodic motion considered 
above would be that of a body which could pass freely through the 
earth. Such a body would vibrate through the centre from surface 
to surface. Calling the diameter of the earth 20919300 feet, we 
would have in this case 

a = 32.17/20919300; 
.-. period = 7ra-i/2 - 3.1410 V20919300/32.17 sec. 
= 42 min. 13.4 sec. 

13. Assuming that the acceleration of a falling body above tlie surface 
of the earth varies inversely as the square of its distance from the oartlTs 
centre, find the velocity and time. 

Let x — the varying, and c the initial, distance of the body from 
the earth's centre ; 

r = the radius of th(^ earth ; 

g = the acceleration due to gravity at its surface ; 

(i = the acceleration due to gravity at the distance x. 

Here ,s = r — .r, and from tlu^ law of fall 

(I : g = f- : .r- ; or ii = gr- /.v'-. 
.-. -d-.v/dr-=-- gr-/x-. (1) 



256 INTEGRAL CALCULUS. 



Since v = when x = c, from (1) we have 

dx ,. /I l\i/2 

Since ^ = when x = c, from (2) we obtain 
1/2 /^x —xdx 



/I l\i/2 






14. Assuming that r, the radius of the earth, is 3962 miles ; that the 
sun is 24,000 r distant from the earth ; and that the moon is 60 r distant ; 
find the time that it would take a hody to fall from the moon to the 
earth, and the velocity, at the earth's surface, of a body falling from the 
sun. The attraction of the moon and the sun, and the resistance of any 
medium, are not to be considered. 

15. A body falls in the air by the force of gravity, the resistance of 
the air varying as the square of the velocity ; determine the velocity on 
the hypothesis that the force of gravity is constant. 

Let b = the resistance when the velocity is unity ; 

and t = the time of falling through the distance s. 

Then h{ds/dt)- = the resistance of the air for any velocity ; 
and g = the acceleration downward due to gravity alone. 

Hence, g — h{ds/dt)'2 = the actual acceleration downward ; 
that is, d^s /dt^ = g -b{ds/ dt)^. (1) 

Integrating (1) and solving for ds/dt, we obtain 
ds jg e^t^^ -1 



x/ 



dt \ b e^t'^bg -^ I 
As t increases, v rapidly approaches the constant value '^g /b. 

16. A body is projected with a velocity, Uq, into a medium which 
resists as the square of the velocity ; determine the velocity and the dis- 
tance after t seconds. 

Let b = the resistance of the medium when the velocity is unity ; 
then b (ds/dt)^ = the resistance for any velocity. 

Hence, d^s /dt^=—b {ds / dtf. (1) 

Integrating (1) and solving for ds/dt, we obtain 

v = ds/dt = Vo/^^ (2) 



APPLICATIONS TO MECHANICS. 



257 



Integrating (2) and solving for s, we obtain 

s = \og{bvot-{- !)/&. 
The velocity decreases rapidly and =0 when s = co. 

17. A body slides without friction down any curve, mn. The accelera- 
tion caused by gravity at any point, P, is g cos BPA, PA being a tangent. 
Find the velocity of the body. m 

Let PA = ds ; then — PD = dij. 
.'. d^s/dt^ = g cos DP A 

= -g-dy/ds. (1) 

Let ?/o be the ordinate of the start- 
ing point on the curve ; then v = 
when y = yo- 

Integrating (1), we obtain 




v = ds/dt = ^2g{yo-y)- (2) 

From (2) it follows that if a body falls from the line ?/ = ?/o to the 
line y = b, the velocity acquired is the same for all curves of descent. 

18. A body falls from the point P along the arc of a cycloid, PO ; 
find the time of descent. 




0) 



(2) 



From (2) of example 17 we have 

V = ds/dt = V2f7(//o — ?/). 
The cciuation of the cycloid referred to OX and OY is 

x = r vers-^ (?//^) + ^^2 r// — ?/-. 
Ilencc, ds — —'^"Ir/y dy. 
Eliminating (?.s betwoen (1) and (2) nnd intogrntiui;-, wo obtain 
t = Vr/f; [tT — vers- ' (2 /////o) ]• 
.*. t = TT^r/g, when // = 0. 
Hence, if a ptMululum swings in tho an- i^f a oyoloid. tlio tinn 
required for one oscillatiiui is 2 tt^ r/g. 

The time of an oscillation bring in(li>|UM>iUMit of \\\c UMii^th of tlu 
arc, tlu> cyt'loidal itiMulnluni is isochronal. 



APPENDIX. 



3^®<C 



FORMULAS FOR REFERENCE. 



Standard Forms. 

log a 



dii = — cosM, or versw. 



6. I cos udu = sinu, or — covers m. 



n + 1 J 

5. I siiun 

. I sec'-^it(i« = taiMt. 11. I tan ?r 

I csc'^udii= — cotu. 12. J ( 

9. I secit tan?t(?(f = sec.?^ 13. j cse?/ 

0. I CSC 21 cot u da = — esc a. 14. j sec u da = loi; tan ( - + ' ) 

/* du 1 , ?^ 1 , u 

i), I , , — -:r:-(.;\n— ' ' or cot ' - • 

^ u- + (f> (r a a a 



du = loiT sec u. 



8. I csc^ K d(t = — cot ?<,. 12. I cot ud II = Uvj: i>in u. 



du = \oiX tan - 



260 APPENDIX. 

r* du 1'- u — a 1, a — u 

16. I — = -— log — ■ — ? or -— log — . 

J u^ — a^ 2a °u-{- a 2a a + u 

17. I : = sm—^-i or— cos— !-• 
J Va2 - m2 a a 

/ , = log (m + Vit^ ± a-). 

Vw^ ± a^ 



18 ' ^^ 



^ „ /• du 1 . ?i 1 - u 

19. I — , = - sec— 1 - ' or esc— i - • 

J wVii--2-a--i a a a a 



20. I — = = vers— 1 -•> or — covers— ^ - • 



du ' ^ u , w 

. ■=^= = vers— 1 -•> or — covers—^ - • 

V 2 ait — m2 a 0^ 



Elementary Principles and Formulas. 

21. I (p{u)du =fu+C, when cZ/ii = <p{u)du. 

22. r((Zu + (Z?/ + d2)= rdu+ TcZ?/ + Cdz. 

23. radu = aCdu. Co = C. 

24. j w cZu = MU — j V du. 

Forms Involving a + bu. 

26. Ct^^, = h flog (« + "") + -nrl • 

J (a + &it)2 &^ L a + &wJ 



FORMS INVOLVING a + buL^. 261 



/yp' du 1 r r/2 

^^+5;^. = ^-5 [a + 6^ ~ 2 a log (a + bu) 



a + 6ti 



„„ . cZu 1 - a + 6it 
29. 1 — — — r-T = log 



) u{a + bu) a '^° u 

/ du 1_ 
u{a+buy^ a{a-hbu) cC^^''^ u 

/du 1 , 6 , f/, 4- i 
~v: — I 7 X = h — log 
w- [a + bu) au a^ u 



30. I ^^^ = — L^_li,-« + ^^ 



Forms Involving a + 61*2. 



32. i — , , 't = — ,;= tan— ^ w a /- •> when a >> and 6 >- : 

33. = — ; log — = ^= ' when a > and 6 < 0. 

2V—ab -Va — uV—b 



34 f *L_ ^ ^^ + — C- 

I (a + bu'^y^ 2 a (a + 6a-) 2 a^ a 



da 



rZ?/, 1 7/ , 2 r — 1 /* (/?r 



Qr r ^^^*' — J[_ 7/ 2r — 1 r__dn__ . 

'^' J ('^ + ?>(6-)'- + ^ " 2?7i {(I + bi(-)'- 'Ira J [a -f bu-)'- ' 

^ Urdu. _n _a /• du 

■ ^ j a + bifi b 6 J rt + 6'f"-^ 

07 r ^'"<?" ^ -^^ 1 _i r ^_ '^" 

*" J ((t + 6(f-^)'' + i 2 7-/>(a, + bu~Y 2 r^J ((? + bu^y 
7 w. {a + bu-) 2 a a + 



39. 



40. 



du 1 /) /' (f» 



<J_n 1 /' du _ 

u~{a-{-bu-) au a J a + bu- 

r du _ 1 r du b r du ^ 

u-{a + buy- ^ ' ~ "./ "'(" + '"'")'' "J 1" "^ /"*■-)'•+» 



262 APPENDIX. 

Forms Involving a + bu^. 
41. j it™(a + 6it")^c?it 

u7n-n + i/a-\-bu»)p + ^ a(m — n + l) /» , , ■, ^ -, 

= TT \ TT^ TT \ TIT I U^~''{(^ + hu»)Pdu; 

6(wp + m + l) b{np + m-\-l)J ^ ' ' 



2t "' + i(a + bun)p anp 

np + m + 1 np + in -\- 1 



42. or YJ^ ^\'' + „^^ Z'Z ^ 1 I w'«(a + bw*)P-'^du] 



43. or 



a{m + 1) a{m + 1) ^ 



it"^ + 1 (a + bu»)p + inp + ?w + n + l/» ^ ,, , ^,_ 
44. or Z/r^ A. i\ f" ^,v. ^ •^ 4- 1 \ — I ^^"' ^^ "^ 6w")^ + 1 (^^^. 



an{p + 1) a7i(p 



Forms Involving au^ + bu + c. 

,^ . du 2 , ^ 2aiL + b 

45. I — —7— \ — = :tan-i 



/ du 
aur + bu + 



c V4 etc — 62 V4ac — 6^' 



1 - 2 aw + 6 — V 62 — 4 dc 

46. = : lo 



47. 



V62 — 4 ac 2 aw + 6 + V 62 — 4 ac 
udu !,,„.,.. 6 /* (?M 



J a2t2 + 6u + c 2 a ° ^ ^ 2 a J 



ait2 + 6it + c 



Forms Involving Va + bu. 

Ao r / , X. ^ 2(2a-36ii)(a +6m)3/2 
48. jwVa + 6M(^M = ^^ TTT^; ^ — 



15 62 



2 (8 a2 - 12 a6u + 15 62^2) {a + 6it)3/2 



49. Cu^^a + budu= ,^,^, 

^ 105 6-^ 

^ __udu__ _ _ 2 (2 g — 6?^) Vg + 6u . 
J Vg + 6w ~ 3 62 

z"* 7t" du _ 2n»Vg + 6it _ 2ng /* u'^—'^du ^ 
J V« + 6m ~ (2w+l)6 (2w + l)6j Vg + 6it 



FORMS INVOLVING Va^ _ 1^2. 263 

CO Z' f^'^ 1 1 Vg +6a — Vg 

52. 1 — ■ = —p log T=r ' when a > ; 

^ uVa-hbu Vg Vg+ 6it + V g 

ro 2 ^ ^ la + bu ^ ^ _ 

53. = , tan— 1 a / 5 when g < 0. 

V3^ \ -g ^ 

/» du _ _ _ Vg + 5u _ {2n — 3)6 /^ tZit 

J ^t'V^T67t~ (n-l)gu»-i (27i-2)aJ uu-iV^TTT^Ti 



. Vg + 6w , ^^/ — r-r- . f* du 



du = 2vg + 6u + g I — 7= 

u J uV^ 



+ 5m 



56. 



Forms Involving Vg^^-u^. 



, ^ — sm-i-- 

Vg2 — %c^ a 

/du ] , ?<, 
— ^^ — log — 
U ^dr- — U^ ^ ^ + Vg--^ — 162 

du Vg''^ — '((2 ^ 



58, 



/ du 
it^ Vg2 — u^ 

59. j Vg'- — 11-^ du = - Vg- — u- + j^ sin— 1 



g-" . , K 
, sni— 1 - 
3 g 



60. I 11- V a- — u'^ du = - (2 u- — a-) Vg^ _ ;,-> + !— sin- 1 - • 
I 8 ^ ' S g 



r^"^~" 7 /":. :> 1 ^T + Vg- — u- 

61. I — du. = V(c- — u- — g log • 

,/ u ' u 

62. I du= sni-i, • 

63. I —-=-== = — - V(r- —)(■'+ , sin- ^ - • 

64. I ; —, = — • 



264 



APPENDIX. 



65. 



66. 



r (a2 - m2)3 /2 du = '^^ (5 a^ - 2 u^) Va2 - ^2 + 

/ U2 c?it _ u 

(a2-M2)-3/2- Va2 - W2 



3 a* . , w 



sm— 1 - • 

a 



67 



/ 



cZi< 



Vu2 ± a- 

68. i — , = - sec-i - 

J uVu^-a^ a a 

69 

du 



Forms Involving Vu2 ± a^. 

log (U + Vu2"±^). 



/du 1 , It 

— , — = - log -^=^== 



70, 



71. 



72. 



^ m2 Vu2 ± (X 

/ du 
u^ Vw2 — a2 



Vzt2 ± ft-^ 



= =F 



2 a-tt^ 2 a-3 a 



VmM^ , 1 , a + Vit2 + a2 



w3 Vu2 + a2 ~ 2a2t^2 2a3 ^ 

73. r Vm2 ± a2 (^w = ^ Vw2 ± a2 ± ^ log {u + Vm2 ± ^^2). 

74. rit2 Vm2 ± a2 du = ^ (2 it2 ± a2) Vu2 ± ^2 _ |- log (it + Vm2 ± a2 

/Vw2 _ a2 ^- a 
dw = V 1(2 — a^ — a cos— ^ - • 
u u 

— du = Vu2 + a2 — a log 

u ^ u 

f 



75 



76 



Vw2 ± a2 V«2 ± a2 
T du = 



+ log (it + Vm2 ± ^2). 



FORMS INVOLVING ^tau-u^ 

\C^ du u 



79. 



, = - Vm2 ± a^ qi — log (u + -yy? ± (J?) 

Vm2 ± a2 ^ ^ 

/ du _ ± u 



80. ' '^'^^ 



81. r(M2±a2)3/2, 



■dw 



^* (2 m2 ± 5 a^) Vw2 ± a2 + ^ log (« + Vu^ ± a^}. 

o o 



82. 



J V2. 



Forms Involving V2 au — 1*2. 



/ 'j<"'cfu _ _ ?t"' — JV2mt — y^-^ (2 ??i — l)a / ^ n"'-U lu 
V2 aw - 162 ?'i wi J V2ait-«2 

f ^ 



84. ' '^'' 



■V2 au — ?/2 ?/? — 1 /• (7»6 



— _ -^ ""' ~ ^^ 4_ ?» — I /• (in 

(2 in- ]) aw" (2 vi - \) a J ^^,. -^ 1 x^aa -u- 

5. I V 2 aiL — Li^ da = — ; — V2 aa — u- H sin- 1 • 

J 2 2 a 

). I w"'V2 



86. i w"'V2tt<(, — u2(i/t 



7/i + 2 //( + 2 } 



87. 



{^ V2 g?/ — u2 dr( 



(2mt-?<2)3/a »< - ;>> r V2(T>< - uh 

(2 /// — ;>) aiC" (2 /yi — \\) (f , ' «"• - * 



266 APPENDIX. 



Forms Involving V± au- -h bu + c, where a > 0. 

88. C / ^^ =^-^log(2a.u + 6 + 2VaVau2 + 5u + c). 

J Vau^ + bu + c Va 



89. I V ait^ + 6m + c c^u 



2 ait + 6 / — „ , , — ; — b^ — 4:ac /^ du 

^av?- + 6it + c -z I , 

8a J Vau2 + 



4(* 8a J Vau2 + 6ii + c 

„^ /» dM 1 ., 2 ait — 

90. I , = — ^sm-i ^ =^- 

J V — ait2 + 6u + c Va V62 + 4ac 

91. j V— ait^ + 6tt + cdw 

2 aw — 6 / ^ , , — ; — , 62 -f- 4 cjc /» cZtt 

- V— av?- + 6w + c H I , 

8a J V- aM2 + 



92. 



4a 8a J V- aM2 + 6m + c 

(?2t 



udu 

V± aM2 + 6it + 

6 /* dw 

2aJ V± ait2 + 6u + c 



Vzb ait2 + 6m + c 6 /* dit 

4- rl "F 



/• 



93. I u V ± ait2 + 6m + c dw 



(±aM2 + 6M + c)3/2 b r n: — . ^ , ■ . 

= ^^ — n: -— I V± att2 + 6m + c ctM. 



^2-J 



Forms Involving Transcendental Functions. 

/M 1 
siii2udM = - — 7 sin 2 It. 
2 4 

/M 1 
cos2 M d(X = - + - sin 2 M. 
2 4 

96. j sin2 u cos2 w(^m=:-^u — Tsin4itj* 

/sec M CSC M cZm = I 
J sm u cos M 



97. I secMcscMdM= i = log tan w. 



TRANSCENDENTAL DIFFERENTIALS. 267 

du 



98. Jsec^«cse^»..=J 
j sin'" It 



siii^ u cos'-^ u 



= tan u — cot u. 



99. i sin'" u cos'' u du 



sin'«-iw cos'^ + Im , m — 1 /* . 

-I , — I sm»^ - ^u cos" u du ; 



m + n m + n 



,^^ sm'^ + iit cos'* — 1 w , n — l . . „ , 

100. = \ 1 \ — I sm'«M cos«-2^4cZu. 



/sin'« u 

,^^ /• . - sin'«— 1^* cosit , m — 1 r* . 

101. I sm'^itdu^ 1 I sin'«- 

J m m J 

-.«r» /• 7 sin w COS'* - 1 ?t , ^ — 1 /* 

102. I cos'»udw = 1 I cos'^-iwd 

J ^^ ^ J 

•,^0 /'sin'" It ., Rin'" + Ut , n — m — 1 /•si: 

103. I du — - 1 — I — 

J COS«U (?l — 1) cos"— Ut 71—1 I CC 



.. r^ .. r COS'* u , cos" + ^ w , -in — n — 2 /"cos" ?< du 

104. I -^ du = — -—-. \ — I — 

J sin'«u (??i — 1) sni'«-i2t m — 1 J sin"'--it 

-.^r- /* dii cosK , m — 2 /* du. 

105. I = — ——. : 1 I -— • 

P sni'"it (?M — 1) sni'" — i It VI — 1 J sm'" — -zt 



^^- /* du _ sin?t , 71 — 2 /* d}i 
lUb. I — 7; r- -\ 7 I ^— 

J cos" it {n — l)cos" — ^tt 7t— 1 f cos" — -it 

tan«xtd(t = --^— I tan"--w(Zit. 

/, cot"-i?t /* . .> 7 
cot" it tx(t = ; I (:oV' — -udu. 
n - 1 J 



108 



109. 



——. = , tan- U \ --- ■ tan - )' it a- > b- ; 

a + b cos u Vfr- — b- \\ a -h b 2 / 



., V/) — a tan ^ 4- ^ ^ -f- (i 
110. = . log = , if ii-<b\ 



V6-^-a'^ V6^r7jt;ui?i-vn^ 



268 APPENDIX. 



111. J»».n„.„ = -u»cos„ + ,„J„«-eos„... 

112. I !(,'" COS u du = w^ sin u — m | Z4'« — i cos u du. 

^-.r. /•sin It , u^ , ti^ 16^ 2^9 

113. j--d„ = ,<-— + _-_ + _.... 

114. r™^d«= — L^itoH+_j^r«>s3i 

J rt'» ?/t — 1 w'»-i m — lj M'»-i 



115. 1 c^it = log u — - — — + 

J u 



du. 



V^ JU^ u^ u^ ^ 

2-[2"^4-[4 6-[68-^'*** 



-.-.« /•cos 16 ^ 1 COS It 1 /•smii ^ 

116. 1 du = r I du. 

J u"^ m — 1 it'«-i m — Ij it'«-i 

117. j It sin-i udu = - [(2m2 — 1) sin-i w + wVl — ^2]. 

^.,„ /• . , ' w«+isin-ii6 1 /^w^ + ^du 

118. I u'^sm— iwdu = — -— - I — , 

J n+l n+lj Vl-u2 



^^^ r , , it« + icos-^w. , 1 pw^ + ^du 

119. I W«COS-lW(Zu = — ; 1-^ r— I , 

J n+l ^^ + 1 J Vl— u2 

^„^ /• , , it« + itan-iM 1 ru^ + '^du 

120. J «» tan-i«du = -^^^^^ ^-^^ J ^^^ • 

121. J„»Iog«d„ = „...[i<f^-^]. 

122. I ii"e«"du = I w'i-ie^^dit. 

J « «J 

123. I — du = • 7 ^ I r du. 

J M« n — 1 ii«-i ^ — Ij w'*~ 

124. I e«« \ogudu = '■ I — du. 

J a aj u 



TKANSCENDENTAL DIFFERENTIALS. 269 

e^" {a sin nu — n cos nu) 



125. I e«" sin nu du ^ , o 

a^ + n^ 



f 

e"" (a cos nu + ?i sin ?im) 



/ 



126. I e'''' cos nil du= ^ , 

a^ + ti' 



Miscellaneous Forms. 



r j a-\- u 
J \b+u 



127. I \ h-^—du 



= V{a + 2i){b + u) + (a — 6)log(Va + M + V6 + ?<). 
To prove formula 127 let 6 + m = z'^. 



128. fyjf:^ du = ^{a-u){b + It) + {a + b) sin- 1 ^|^ 



129. r-=^^_=2cot-iJ^^^ = 2sin-iJ;^^ 

130. I _- =r — log . 

J uVm» + a- ^'^ Va"^ + le' + a 
/ u Vu" — a- 



131. I :-r—:— — =: SCC— ^ 

an a 



SHORT COURSE IN THE CALCULUS. 

To those who wish to give in Taylor's Calculus a short coui-se 
including the fundamental principles, problems, methods, and 
applications of the Calculus, the following suggestions may 
prove helpful : 

PART I. 

Chapter I. Take it all thoroughly. 

Chapter II. Omit exs. 28-34, pp. 18, 19 ; exs. 6, 8, 12, 13, 

14, 25-30, pp. 19, 20 ; exs. 8-10, pp. 21, 22 ; exs. 19-23, 

pp. 28, 29; exs. 23-29, p. 32; exs. 18-23, pp. 35, 36 ; exs. 

1-30, pp. 38, 39. 
Chapter III. Omit all after p. 44 except § 74. 
Chapter IV. Omit all after ex. 15, p. 60, except the defini- 
tion in § 82. 
Chapter Y. Omit exs. 9-14, p. 67; exs. 8-11, p. 68 ; all the 

chapter after ex. 9, p. 69. 
Chapter YI. Head the proof of Taylor's theorem with the 

class, but do not require its reproduction. Omit Cors. 

2-4, p. 74; §§ 93, 96, 99; the proofs of convergency in 

§§ 94, 95, 97, 98 ; exs. 11-20, p. 81. 
Chapter YII. Omit exs. 12-19 and 22, p. 87 ; exs. 14-16, 

p. 90 ; exs. 19-24, pp. 91, 92. 
Chapter YIII. Omit § 110; exs. 8-11, p. 97 ; § 116 and 

examples ; exs. 7-9, p. 102. 

Chapter IX. Omit exs. 8-13, pp. 106, 107 ; exs. 1-8, p. 111. 

Chapter X. Omit it all. 

270 



SHORT COURSE IN THE CALCULUS 271 

Chapter XI. Omit pp. 118-126. 

Chapter XII. Omit exs. 7-9, p. 130 ; exs. 9-12, p. 132 ; § 149 
and examples ; § 153 and examples. Bead curve tracing 
with tlie class so that the most important curves are made 
familiar. 

PART II. 

Chapter I. Omit exs. 39-48, p. 150 ; exs. 57-59, p. 151 ; 

exs. 22-24, p. 153 ; exs. 19-24, pp. 155-156. 
Chapter II. Omit exs. 4-6, p. 160 ; § 168, p. 166. 
Chapter III. Omit exs. 7-9, p. 169 ; exs. 6-8, p. 170 ; exs. 

8-12, p. 172; exs. 3-6, p. 173. 
Chapter IV. Omit exs. 6, 10, p. 175 ; exs. 7-9, p. 177 ; exs. 

8-11, p. 180. 
Chapter V. Omit exs. 17-26, p. 184; exs. 10, 14, 19-21, 

25-31, pp. 188, 189. 
Chapter YI. Omit exs. 11-15, p. 192; exs. 10-13, p. 194; 

exs. 5-7, p. 195 ; §§ 190, 191, and examples, pp. 196, 197 ; 

§ 192, and examples, pp. 198, 199 ; exs. 5-9, p. 200 ; § 194 

and examples, p. 201 ; exs. 7-9, p. 202. 
Chapter VII. Omit exs. 5, 7, p. 204 ; § 198 ; exs. 3, 4, p. 207 ; 

§ 200 and examples ; exs. 7-13, p. 210 ; exs. 6-9, p. 211 ; 

exs. 5-S and 13-17, pp. 214, 215; § 205 and examples. 
Chapter VIII. Omit all after ex. 3, p. 221. 

To gain an idea of limits and iniinitesimals as used in the 
(■alculus, read (chapter III in l*art I and Chapter IX in 
Tart II. 

It' a course still shorter than that outlined above is desired, 
omit all of (^hapters VII XIl in Tart I, all of Chapter VIU 
ill Part II, ami more exainph^s in ilu^ other chai^tiM's. 



